# FACTORS

Any number which is an exact divisor of a given number is called a factor of the given number.
For example factor of 6 are 1, 2, 3 and 6
Important results :
(a) 1 is a factor of every number.
(b) Every number is a factor of itself.
(c) Every factor of a number is always equal to or less than the number.
(d) Every number has a finite number of factors.

## MULTIPLES

Just as 1, 2, 3 and 6 are factors 6, we say that 6 is multiple of 1, 2, 3 and 6
A number is a multiple of each of its factors
Important results :
(a) Every number is a multiple of itself.
(b) Every multiple of a number is equal to or greater than the number
(c) Every number has as infinite number of multiples.

### TYPES OF NUMBERS

(a) Even Number : A number which is exactly divisible by 2 is called an even number.

(b) Odd Number : A number which is not exactly divisible by 2 is called an odd number.
Example of odd numbers are : 1, 13, 15, 25, 29, ...........
(c) Prime Numbers : A natural number greater than 1, which has no factors except 1 and itself is called a prime number.
Examples of prime numbers are : 2, 3, 5, 11, 13, 17, ........

### NOTE:

Every even number greater than 4 can be expressed as a sum of two odd prime numbers,
e.g., 6 = 3 + 3; 18 = 5 + 13; 44 = 13 + 31.

(d) Composite Numbers : A number is composite if it has at least one factor other than 1 and itself.
Example of composite numbers are 4, 6, 8, 9, 10, 12, 14,........

### NOTE:

1. 1 is neither prime nor composite.
2. Every natural number except 1 is, either a prime number or a composite number.
3. 2 is the only prime number which is even. All other prime numbers are odd.

(e) Twin primes : Pairs of prime numbers that have a difference of 2 are called twin primes.
Example of twin primes are : (3, 5), (5, 7), (11, 13), (17, 19),........

(f) Perfect Numbers : If the sum of all the factors of a number is twice the number, then number is called a perfect number. For example, 6 is a perfect number since the factors of 6 are 1, 2, 3, 6 and their sum 1 + 2 + 3 + 6 = 2 × 6.

(g) Coprime Number : Two numbers are said to be coprime if they do not have a common factor other than 1.
Examples of coprime numbers are : (8, 15); (5, 9); (2, 11)

### NOTE

1. Two prime numbers are always coprime.
2. Two coprime numbers need not be both prime numbers.

(h) Prime Triplet : A set of successive prime numbers differing by 2 is called a prime triplet. The only example of a prime triplet is (3, 5, 7).
Divisibility Tests For 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

(a) Test of divisibility by 2
A number is divisible by 2 if its one’s digit is an even digit. For example 206 is divisible by 2 because 6 is even.

(b) Test of divisbility by 4
A number is divisible by 4 if the number formed by its last two digits (One’s and Ten’s) is divisible by 4. For example 312 is divisble by 4 because 12 is divible by 4.

(c) Test of divisibility by 8
A number is divisible by 8 if the
...
formed by its last three digits (One’s Ten’s Hundred’s) is divisible by 8. For example 8864 is divible by 8 because 864 is divisible by 8.

(d) Test of divisibility by 5
A number is divisible by 5 if its one’s digit is 0 or 5. For example 3075 is divisible by 5 because its one digit is 5.

(e) Test of divisibility by 10
A number is divisible by 10 if its one’s digit is 0.
For example 3760 is divisible by 10 because its one’s digit is 0.

(f) Test of divisibility by 3
A number is divisible by 3 if the sum of its digits is divisible by 3.
For example 567 is divisible by 3 because sum of digit of 567 (5 + 6 + 7 = 18) is divisible by 3.

(g) Test of divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
For example 4563 is divisible by 9 because sum of digit of 4563 (4 + 5 + 6 + 3 = 18) is divisible by 9

(h) Test of divisibility by 6
A number is divisible by 6 if it is divisible by both 2 and 3.
For example 456 is divisible by 6 beacuse its one’s digit is even so it is divisible by 2, and the sum of digit of 456(4 + 5 + 6 = 15) is divisible by 3. So 456 is divisible by 2 and 3 both, so we can say that it is divisible by 6.

(i) Test of divisibility by 11
A number is divisible by 11 if the difference of the sums of the digits at the alternate places is zero or divisible by 11.
For example 45672 is divisible by 11 because sum of digit 4, 6 and 2 at alternate palce 4 + 6 + 2 = 12 sum of digit 5, 7 at alternate place 5 + 7 = 12
So their difference (12 – 12 = 0) is zero, so 45672 is divisible by 11.

### NOTE :

A given number will be divisible by any other number say, n, if it is divisible by the coprime factors of n.
For example, 9624 is divisible by 12, because it is divisible by 4 and 3 (the coprime factors of 12).

### SIEVE OF ERATOSTHENES

How can we list all the prime number between say, 1 and 100? Eratosthenes (274 B.C. – 194 B.C.), a Greek mathematician, gave a simple method to mark out primes. His method is knows as the Sieve of Eratosthenes.
We first list the number upto 100, except 1

### SIEVE OF ERATOSTHENES

1. Begin with 2 which is prime. So keep it but cross out all its multiples.
2. Next, the number 3 is prime. Thus we keep it but cross out all its multiples. Some of these numbers have already been crossed out.
3. The next number not crossed out is 5. It is also prime. So keep it and cross out all its multiples.
4. Continue this process keeping only the primes and striking off their multiples until we cannot strike off any more numbers.

Thus the prime numbers from 1 to 100 are :
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
Eratosthenes, probably, made holes in the paper instead of crossing out the numbers. Therefore, his paper must have look like a sieve. That is why perhaps this method is known as sieve method.

Observations. Some observations about prime numbers are :
(i) 2 is the smallest prime number.
(ii) All prime numbers (except 2) are odd numbers.
(iii) The number of primes is unlimited.
(iv) Both the numbers 13 and 31 have the same digits and are prime. Other such numbers between 1 and 100 are : 17, 71; 37, 73 and 79, 97.
(v) Every odd prime number can be expressed as a product of primes plus 1.

For example
3 = 2 + 1; 5 = 2 × 2 + 1
7 = 2 × 3 + 1; 43 = 2 × 3 × 7 + 1

### TO FIND PRIME NUMBERS BETWEEN 100 AND 400

We know that 20 × 20 = 40
So we adopt the following rule :

Rule : The given number will be prime if it is not divisible by any prime number less than 20.

Ex. Find out whether 397 is a prime number or not.
Sol. Because 397 < 400, we check whether 397 is divisible by any prime number less than 20.
The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, 19. Let us test the divisibility of 397 by each of them. 397 is not divisible by 2 because the digit in the ones place is odd.
397 is not divisible by 3 because 3 + 9 + 7 = 19, but 19 is not divisible by 3. 397 is not divisible by 5 because the digit in the ones place is neither 5 nor 0, 397 is not divisible by 7 because 397 ¸ 7 gives quotient 56 and remainder 5. 397 is not divisible by 11 because the difference of the sum of the digits at the alternate places is 1 which is not divisible by 11.

Now 397 is not divisible by 13 because 397 ¸ 13 gives quotient 30 and remainder 7.
397 is not divisible by 17 because 397 ¸ 17 gives quotient 23 and remainder 6.
397 is not divisible by 19 because 397 ¸ 19 gives quotient 20 and remainder 7.
Since 397 is not divisible by any prime number less than 20, so 397 is a prime number.
Prime Factorisation or Complete Factorisation
A factorisation is prime if all the factors are prime.

For example prime factorisation of 120 is
120 = 2 × 2 × 2 × 3 × 5 = 23 × 3 × 5
The prime factorisation is unique
60 = 60 × 1 = 30 × 2
= 15 × 2 × 2 = 5 × 3 × 2 × 2

In writing 60 as 5 × 3 × 2 × 2, we see that each of the factors of 60 is a prime number. When we factorise a number into prime factors, we say that we have written the prime factorisation or complete factorisation of the number.
Fundamental Theorem Arithmetic
Every composite number can be factorised into prime in only one way, except, for the order of primes.

Common Factors
Numbers which exctly divide two or more numbers are called their common factors.
All factors of 18 are : 1, 2, 3, 6, 9, 18
All factors of 24 are : 1, 2, 3, 4, 6, 8, 12, 24
Common factors of 18, 24 are : 1, 2, 3, 6 as they divide both 18 and 24.

### HIGHEST COMMON FACTOR (H.C.F.)

To find highest common factor (H.C.F.) or greatest common divisor (G.C.D.) of two or more numbers, we adopt the folllowing method. Let us find H.C.F. of two numbers say 16 and 40.
All possible factors of 16 are : 1, 2, 4, 8, 16.
All prossible factors of 40 are : 1, 2, 4, 5, 8, 10, 20, 40
Now the common factors of 16 and 40 are : 1, 2, 4, 8.
The highest of all these common factors is 8.
Finding HCF by Prime Factorisation

### STEP 1.

Find the prime factorisation of the given numbers.

### STEP 2.

Find the common factors and circle them.

### STEP 3.

Multiply the common factors to get HCF.
Let us now find the HCF of 72, 64 and 48.

### STEP 1.

Find the prime factorisation of each the numbers.

We can factorise the numbers as follows :

72 = 2 × 2 × 2 × 3 × 3
64 = 2 × 2 × 2 × 2 × 2 × 2
48 = 2 × 2 × 2 × 2 × 3

### STEP 2.

Take the common factors in all the given numbers.
72 = 2 × 2 × 2 × 3 × 3
64 = 2 × 2 × 2 × 2 × 2 × 2
48 = 2 × 2 × 2 × 2 × 3

### STEP 3.

Multiple the common factor to get the HCF.
The HCF of 72, 64 and 48 = 2 × 2 × 2 = 8
Finding HCF using Division by Common Factors
Divide all the three numbers by any factor common to all of them. If there are till any common factors, again divide the quotients by them and keep dividing until there is no common factor for all three of them. The product of these common factors will give the highest common factor (HCF) of these number.

### H.C.F. BY DIVISION METHOD

In this method, we divide the greater number by the smaller number. Then the remainder is treated as divisor and the first divisor as dividend. We continue this operation till we get the remainder zero. The last divisor is the H.C.F. of the two given number. We illustrate this method by the following exmples.

Ex. Find the H.C.F. of 345 and 506.

Sol

The last divisor is 23.
\ H.C.F. of 345 and 506 is 23.
To find the H.C.F. of three number, first we find the H.C.F. of any two numbers. Then treating this H.C.F. as one number and third number as another number, we find their H.C.F. by the method stated above. The H.C.F. so found will be the H.C.F. of the three numbers.

Ex. Find the H.C.F. of 219, 2628, 2190 and 8833.
Sol. First we find the H.C.F. of 219 and 2628.

Now we find the H.C.F. of 219 and 2190.

H.C.F. of 219, 2628 and 2190 = 219.
Now we find the H.C.F. of 219 and 8833.

Hence the H.C.F. of 219, 2628, 2190 and 8833 is 73.

Ex. Find the greatest numberthat will divide 18 and 48 without leaving a remainder.

Sol. Required number is the HCF of 18 and 48.
HCF of 18 and 48 is 6
18 = 2 × 3 × 3
48 = 2 × 2 × 2 × 2 × 3
Required HCF = 2 × 3
the greatest number that will divide 18 and 48 without leaving a remainder is 6.

Ex. Find the greatest number which divides 43 and 91 leaving a remainder 7 in each caes.

Sol. It is given that the required number when divides 43 and 91, the remainder is 7 in each case. This means that 43 – 7 = 36 and 91 – 7 = 84 are completely divisible by required number.
Also, the required number is the greatest number satisfying the above property.
\ It is the HCF of 36 and 84.
36 = 2 × 2 × 3 × 3
84 = 2 × 2 × 3 × 7
\ Required HCF = 2 × 2 × 3 = 12
Hence, the required number = 12

Ex. Find the largest number that will divide 20, 57 and 85 leaving remainders 2, 3 and 4 respecitively.

Sol. Clearly, the required number is the HCF of the number 20 – 2 = 18, 57 – 3 = 54 and 85 – 4 = 81.
18 = 2 × 3 × 3
54 = 2 × 3 × 3 × 3
81 = 3 × 3 × 3 × 3
\ Required HCF = 3 × 3 = 9
Hence, the required number = 9.

Ex. The length, breadth and height of a room are 8m 25cm, 6m 75cm and 4m 50cm resprctively. Determine the longest rod which can measure the three dimensions of the room exactly.

Sol. We have, 8m 25cm = 825cm, 6m 75cm = 675cm 4m 50cm = 450cm
Length of the longest rod in cm is the HCF of 825, 675 and 450.
825 = 3 × 5 × 5 × 11
675 = 3× 3 × 3 × 5 × 5
450 = 2 × 3 × 3 × 5 × 5
\ HCF 825, 675 and 450 = 3 × 5 × 5 = 75
Hence, the required length of rod = 75.

Ex. A rectangular courtyard 3.78 metres long and 5.25 metres wide is to be paved exactly with square tiles, all of the same size. What is the largest size of the tile which could be used for the purpose?

Sol. Largest size of the tile = H.C.F. of 378 cm and 525 cm = 21 cm.
Common Multiple
Number which are exactly divisible by two or more numbers are called their common multiples.
Let us find the common multiple of 12 and 18.
Multiples of 12 are 12, 24, 36, 48, 60, 72 .....
Multiples of 18 are 18, 36, 54, 72 .....
The common multiple of 12 and 18 are 36, 72 ....

### LEAST COMMON MULTIPLE (L.C.M.)

To find LCM of two or more number we adopt the following steps :
(i) Find the multiples of the given numbers.
(ii) Select their common multiples
(iii) Take the smallest of the above common multiples
Let us find the LCM of 6 and 8
The multiple of 6 are 6, 12, 18, 24, 30, 36, 42, 48, .....
The multiple of 8 are 8, 16, 24, 32, 40, 48,.....
So common multiple of 6, 8 are 24, 48,.....
The lowest common multiple of 6 and 8 is 24
Hence 24 is the LCM of 6 and 8
Finding LCM by Prime Factorisation

### STEP 1.

Find the prime factorisation of the given numbers.

### STEP 2.

LCM is the product of all the prime factors with greatest powers.
Ex. Find the LCM of 84 and 96.
Sol. 84 = 2 × 2 × 3 × 7 = 22 × 3 × 7
96 = 2 × 2 × 2 × 2 × 2 × 3 = 25 × 3
LCM of 84,96 is 25 × 3 × 7 = 672

Finding LCM by common Division

### STEP 1.

Write the given numbers in a row separated by commas.

### STEP 2.

Divide these numbers by the least prime numbers which divides at least one of the given numbers.

### STEP 3.

Write the quotients and the numbers that are not divisible by the prime numbers in the second row. Then repeat Steps 2 and 3 with the rows and continue till the numbers in a row are all 1.
STEP 4.
The LCM is found out by multiplying all the prime divisors and quotients other than 1.

Ex. Find the L.C.M. of 28, 36, 45 and 60.

Sol.

L.C.M. = 2 × 2 × 3 × 3 × 5 × 7 = 1260
Relationship between HCF and LCM
Let us take two numbers, say 16 and 24.
The HCF of 16 and 24 is 8.
The LCM of 16 and 24 is 48.
Since 8 is factor of 48, so we can say that HCF of the numbers is a factor of their LCM.
Product of HCF and LCM = 8 × 48 = 384
Product of Numbers = 16 × 24 = 384
So we can say that the product of two numbers is equal to the product of their HCF and LCM.
Let a and b are two numbers then a × b = HCF × LCM

Ex. The HCF of two number is 29 and their LCM is 1160.
If one of the number is 145, find the other.

Sol. We know that
Product of the number = Product of their HCF and LCM
Required No. × 145 = 29 × 1160
Required No. =  = 232

### Properties of HCF and LCM

1. The HCF of 6 and 10 is 2. So 2 is a factor of both 6 and 10. Also, 2 is the smallest amongst 2, 6 and 10.

2. The LCM of 6 and 10 is 30. 30 is a multiple of both 6 and 10. Also, 30 > 10 and 6, i.e., it is the greatest amongst 6, 10 and 30.

3. Consider two numbers 36 and 39.
Now, 35 = 1 × 5 × 7
39 = 1 × 3 × 13
Common factor = 1
\ 35 and 39 are co-prime
HCF of 35 and 39 = 1
Thus HCF of two or more co-prime numbers is 1.

4. Again consider 35 and 39 which are coprime.
LCM of 35 and 39 = 3 × 5 × 7 × 13 = 35 × 39
Thus the LCM of co-prime numbers = the product of the co-primes.

5. HCF of 6, 10 = 2
LCM of 6, 10 = 30
Also, 30 = 2 × 15 = 2 × 3 × 5
i.e., 2 is a factor of LCM.
Thus, HCF is a factor of LCM. In other words, LCM is a multiple of HCF.

6. 2 and 3 are prime numbers.
HCF of 2 and 3 is 1.
HCF of two or more prime numbers is 1.
Some More Divisibility Rules

Let us observe a few more rules about the divisibility of numbers.

(i) One of the factor of 18 is 9. A factor of 9 is 3. Is 3 a factor of 18? Yes it is. Take any other factor of 18, say 6. Now, 2 is a factor of 6 and it also divides 18. Check this for the other factors of 18. Consider 24. It is divisible by 8 and the factors of 8 i.e., 1, 2, 4 and 8 also divide 24.
So, we may say that if a number is divisible by another number then it is divisible by each of the factors of that number.

(ii) The number 80 is divisible by 4 and 5. It is also divisible by 4 × 5 = 20, and 4 and 5 are co-primes.
Similarly, 60 is divisible by 3 and 5 which are coprime. 60 is also divisible by 3 × 5 = 15,
If a number is divisible by two co-prime numbers then it is divisible by their product also.

(iii) The numbers 16 and 20 are both divisible by 4. The number 16 + 20 = 36 is also divisible by 4.
If two given numbers are divisible by a number, then their sum is also divisible by that number.

(iv) The number 35 and 20 are both divisible by 5. Their difference 35 – 20 = 15 also divisible by 5.
If two given number are divisible by a number, then their difference is also divisible by that number.

Ex. Find the least number with when divided by 6, 7, 8, 9 and 10 leaves reaminder 1.

Sol. As the remainder is same
Required number = LCM of divisors + Remainder = LCM (6, 7, 8, 9, 10) + 1
= 2520 + 1 = 2521

Ex. Find the least number which when decreased by 7 is exactly divisible by 12, 16, 18, 21 and 28.

Sol.

LCM = 2 × 7 × 3 × 2 × 4 × 3 = 1008
Required number = 1008 + 7 = 1015.
Ex. When 21 is added to a number it is diveded exactily by 3, 8, 9, 12, 16 and 18. How many such numbers exist? Find the least of them.

Sol. We know that the least number divisible by 3, 8, 9, 12, 16 and 18 is their LCM.
Therefore, the required number must be 21 less then their LCM

LCM = 2 × 2 × 2 × 3 × 3  × 2 = 144
Hence, the required number = (144 – 21) = 123
There exists many such numbers (i.e., all the multiples of 123) and least of them is 123.

Ex. In a morning walk four boys steps off together. Their steps measure 70 cm, 65 cm, 75 cm and 80 cm respectively. At what distance from the starting point will they step off together again?

Sol. The distance covered by each one of them is required to be same and minimum both. The required minimum distance each should walk must be the LCM of their steps in cm.

LCM = 2 × 5 × 7 × 13 × 15 × 8 = 109200
They will step off together again after a distance of 109200 cm = 1092 m.

OPERATOR PRECEDENCE
Generally, the order in which we perform operations sequentially from left to right is : division, multiplication, additions & subtraction. This order is expressed in short as 'DMAS' where 'D' stands for division, 'M' for multiplication, 'A' for addition and, 'S' for subtraction.
Ex. Simplify : (– 20) + (– 8) ¸ (– 2) × 3

Sol. We have,
(–20) + (– 8) ¸ (– 2) × 3
= (– 20) + 4 × 3
= (– 20) + 12
= – 8.

### USE OF BRACKETS

In order to simipify expression involving more than one brackets, we use the following steps.

### STEP 1.

See whether the given expression contains a vinculum or not. If a vinculum is present, then perform operations under it. Otherwise go to next step.

### STEP 2.

See the innermost bracket and perform operations within it.

### STEP 3.

Remove the innermost bracket by using following rules :
Rule 1 : If a bracket is preceded by a plus sign, remove it by writing its terms as they are.
Rule 2 : If a bracket is preceded by minus sign, change positive signs within it to negative and
vice-versa.
Rule 3 : If there is no sign between a number and a grouping symbol, then it means multiplication.
Rule 4 : If there is a number before some brackets then we multiply the number inside the bracktes with the number outside the brackets.

### STEP 4.

See the next innermost bracket and perform operations within it. Remove the second
innermost bracket by using the rules given in step III. Continue this process till all the brackets are removed.
Ex. Simiplify

(i) 39 – [23 – {29 – (17 – )}]
(ii) 15 – (–3) {4 – } ¸ [3 {5 + (– 3) × (– 6)}]

Sol. (i) 39 – [23 – {29 – (17 – 6)}]
= 39 – [23 – (29 – 11)]
= 39 – [23 – 18]
= 39 – 5
= 34

(ii) 15 – (– 3) {4 – 4} ¸ [3 {5 + 18}]
15 – (– 3) × 0 ¸ 89
15 – (– 3) × 0 = 15

### OBJECTIVE TYPE

Q.1 The HCF of two numbers is 11 and their LCM is 7700. If one of the numbers is 275, then the other is
(A) 279         (B) 283          (C) 308           (D) 318

Q.2 Product of two co-prime numbers is 117. Their LCM should be
(A) 1                                    (B) 117
(C) equal to their HCF        (D) cannot be calculated

Q.3 The LCM and HCF of two numbers are 4125 and 25 respectively. One number is 375. Find by how much is the second number less than the first?
(A) 100           (B) 50            (C) 75             (D) 25

Q.4 Three city tour buses leave the bus stop at 9.00 AM. Bus A returns every 30 minutes, but B returns every 20 minutes and Bus C returns every 45 minutes.
What is the next time, the buses will all return at the same time to the bus stop.
(A) 1:00 PM           (B) 12 noon             (C) 7:00 PM               (D) 11:30 PM

Q.5 The least number of 5-digits which is exactly divisible by 16, 24, 36 and 54 is
(A) 10638               (B) 10368                (C) 13068                   (D) 1084

Q.6 Find the greatest number of 5-digit which when divided by 3, 5, 8 and 12 will have 2 as reminder
(A) 99999               (B) 99958                (C) 99960                   (D) 99962

Q.7 The length and breadth of a room are 18 m and 28 m respectively. What is the greatest length of the side of a square tile required for pairing the floor of the room (No space is to be left in the room without a tile)
(A) 2.4 m                (B) 4 m                    (C) 2 m                       (D) 3.2 m

Q.8 The prime factorisation of 54 is :
(A) 2 × 2 × 3 × 3         (B) 2 × 2 × 2 × 3             (C) 2 × 3 × 3 × 3             (D) 2 × 27

Q.9 Which of the following is a pair of twin primes?
(A) (7, 9)                     (B) (17, 19)                     (C) (51, 53)                     (D) (31, 33)

Q.10 The only even prime number is :
(A) 2                           (B) 3                                (C) 4                                (D) 0

Q.11 LCM of 15, 20 and 30 is :
(A) 50                         (B) 20                              (C) 60                              (D) 15

Q.12 The smallest digit to make the number 5703_2 divisible by 4 is :
(A) 2                           (B) 4                                (C) 8                                (D) 1

Q.13 The largest 3-digit number which is exactly divisible by 3 is :
(A) 998                       (B) 992                            (C) 999                            (D) None of these

Q.14 Which of the following is a pair of co-prime?
(A) (55, 57)                 (B) (46, 50)                     (C) (72, 78)                     (D) none of these

Q.15 Which of the following is a composite number ?
(A) 1                           (B) 2                                (C) 5                                (D) 8

### Simplify (1 – 6)

Q.1 8 – (4 × 2) ¸ 8.

Q.2 {3 + (4 × 5) ¸ 2 – 6} ¸ 7.

Q.3 2 – [3 – {6 – (5 – )}]

Q.4 73 of [45 – {6 × 7 + (23 – 4 of 5)}]

Q.5 {5(18 ¸ )– 30} + 20 × 10 ¸ 5

Q.6 What is the value of 64 ¸ 8 ¸ 4 ¸ 2?

Q.7 List all the factors of :

(i) 23 (ii) 48 (iii) 168

Q.8 Write the following :

(i) The first 3-digit even multiple of 7
(ii) Odd multiples of 17, less than 100
(iii) Multiples of 5 between 52 and 76.

Q.9 Write the seven consecutive composite numbers less than 100.

Q.10 Replace the star(*) by the smallest number so that

(i) 78 * 964 many be divisible by 9.
(ii) 75 * may be divisible by 9.
(iii) 2 * 345 may be divisible by 3.

Q.11 What least number should be subtracted from 26492518 so that the resulting number is divisible by 3, but not by 9?

Q.12 Three different tankes contain 496 litres, 403 litres and 713 litres of milk. Find the maximum capacity of a container that can measure the milk of any tanker an exact number of times.

Q.13 Find the largest number that divides 220, 313 and 716 leaving remainders 3 in each case.

Q.14 Find the largest number that will divide 623, 729 and 841 leaving remainders 3, 9 and 1 respectively.

Q.15 Find the LCM of the following by division method :

(i) 20, 25, 30, 50
(ii) 9, 12, 18, 24, 27
(iii) 22, 54, 108, 135, 198

Q.16 Find the least number which when divided by 12, 16, 24 and 36 leaves a remainder 7 ?

Q.17 In a walking competition, three person step off together. Their steps measure 85 cm, 90 cm and 80 cm respectively. At what distance from the starting point will they again step off together?

Q.18 Find the greatest number of 4-digit exactly divisible by 12, 16, 24, 28 and 36.

Q.19 Write all the prime numbers between :

(a) 5 and 35 (b) 70 and 100 (c) 40 and 80 (d) 77 and 158
Q.20 Can a composite number be odd ? If yes, write the smallest odd composite number.

Q.21 (a) Is there any natural number having no factor at all ?

(b) Find all the number having exactly one factor.
(c) Find numbers between 1 and 100 having exactly three factors.

Q.22 Express each of the following numbers as sum of two odd primes :
(a) 36              (b) 42                  (c) 84                      (d) 98

In questions 1 to 38 out of the four options, only one is correct. Write the correct answer.

1. The product of the place values of two 2's in 428721 is
(A) 4                       (B) 40000
(C) 400000             (D) 40000000

2. is the same as
(A) 3794                 (B) 37940
(C) 37904               (D) 379409

3. If 1 is added to the greatest 7-digit number, it will be equal to
(A) 10 thousand      (B) 1 lakh
(C) 10 lakh              (D) 1 crore

4. The expanded form of the number 9578 is
(A)
(B)
(C)
(D)

5. When rounded off to nearest thoudands, the number 85642 is
(A) 85600                  (B) 85700
(C) 85000                  (D) 86000

6. The largest 4-digit number, using any one digit twice, from digits 5, 9,2 and 6
(A) 9652                    (B) 9562
(C) 9659                    (D) 9965

7. In Indian system of numeration, the number 58695376 is  written as
(A) 58,69,53,76        (B) 58,695,376
(C) 5,86,95,376        (D) 586,95,376

8. One million is equal to
(A) 1 Lakh                (B) 10 Lakh
(C) 1 crore                (D) 10 crore

9. The greatest number which on rounding off to nearest thousands gives 5000, is
(A) 5001                    (B) 5559
(C) 5999                    (D) 5499

10. Keeping the place of 6 in the number 6350947 same, the smallest number obtained by rear- ranging other digits is
(A) 6975430               (B) 6043579
(C) 6034579               (D) 6034759

11. Which of the following numbers in Roman nu- merals is incorrect ?
(A) LXXX                 (B) LXX
(C) LX                       (D) LLX

12. The largest 5-digit number having three differ- ent digits is
(A) 98978                  (B) 99897
(C) 99987                  (D) 98799

13. The smallest 4-digit number havein gthree dif- ferent digits is
(A) 1102                    (B) 1012
(C) 1020                    (D) 1002

14. Number of whole numbers between 38 and 68
(A) 31                        (B) 30
(C) 29                        (D) 28

15. The product of successor and predecessor of 999 is
(A) 999000 (B) 998000
(C) 989000 (D) 1998

16. THe product of a non-zero whole number and its successor is always
(A) an even number             (B) An old number
(C) a prime                           (D) divisible by 3

17. A whole number is added to 25 and the same number is subtracted from 25. The sum of the resultanting number
(A) 0                                     (B) 25
(C) 50                                   (D) 75

18. WHich of the following is not true?
(A) (7 + 8) + 9 = 7 + (8 + 9)
(B) (7 x 8 ) x 9 = 7 x (8 x 9)
(C) 7 + 8 x 9 = (7 + 8) + (7 + 9)
(D) 7 x (8 + 9) = (7 x 8) + (7 x 9)

19. By using dot(.) patterns, which of the follow- ing numbers can be arranged in all the three ways namely a line, a triangle and a rectangle?
(A) 9                         (B) 10
(C) 11                       (D) 12

20. WHich of the following statements is not true?
(A) Both addition and multiplication of whole numbers.
(B) Zero is the identity for multiplication of whole numbers
(C) Addition and multiplication both are com mutative for whole numbers.
(D) Multiplication is distributive over addition for whole numbers.

21. WHich of the following statements is not true?
(A) 0 + 0 = 0             (B) 0 -0 = 0
(C) 0 x 0 = 0             (D)

22. The predecessor of 1 lakh is
(A) 99000              (B) 99999
(C) 999999            (D) 100001

23. The successor of 1 million is
(A) 2 millions         (B) 1000001
(C) 100001             (D) 10001

24. Number of even numbers btween 58  and 80 is
(A) 10                      (B) 11
(C) 12                      (D) 13

25. Sum of the number of primes between 16 to 80 and 90 to 100 is
(A) 20                     (B) 18
(C) 17                     (D) 16

26. Which of the following statements is not true?
(A) The HCF of two distinct prime numbers is 1
(B) THe HCF of two co-prime numbers is 1
(C) The HCF of two consecutive even num- bers is 2
(D) THe HCF of an even and an odd number is even.

27. The number of distinct prime factors of the largest 4-digit number is
(A) 2                       (B) 3
(C) 5                       (D) 11

28. The number of distinct prime factor s of the smallest 5 -digit number
(A) 2                      (B) 4
(C) 6                      (D) 8

29. If the number 7254*98 is divisible by 22, the digit at * is
(A) 1                      (B) 2
(C) 6                      (D) 0

30. The largest number which alwasy divides the sum of any pair of consecutive odd numbers is
(A) 2                      (B) 4
(C) 6                      (D) 8

31. A number is divisible by 5 and 6. It may not be divisible by
(A) 10                   (B) 15
(C) 30                   (D) 60

32. The sum of the prime factor of 1729 is
(A) 13                  (B) 19
(C) 32                  (D) 39

33. The greatest number which always divides the product of the predecessor and successor of an odd natural number other than 1 is
(A) 6                    (B) 4
(C) 16                  (D) 8

34. The number of common prime factor of 75, 60, 105 is
(A) 2                    (B) 3
(C) 4                    (D) 5

35. Which of the following pairs is not coprime?
(A) 8,10               (B) 11, 12
(C) 1, 3                (D) 31, 33

36. Which of the following numbers is divisible by 11
(A) 1011011       (B) 1111111
(C) 22222222     (D) 3333333

37. LCM of 10, 15 and 20 is
(A) 30                 (B) 60
(C) 90                 (D) 180

38. LCM of two numbers is 180. Then which of the following is not the HCF of the numbers?
(A) 45                   (B) 60
(C) 75                   (D) 90

### In question 39 to 98 state whether the given statements are true (T) or false(F)

39. In Roman numeration, a sysmbol is not repeated more than three

40. In Roman numberation , if a symbol is repeated, its value is multiplied as many times as it oc- curs.

41. 5555 =   5 x 1000 + 5 x 100 + 5 x 10 + 5 x 1

42. 39746= 3 x 10000 + 9 x 1000 + 7 x 100 + 4 x 10 + 6

43. 82546 = 8 x 1000 + 2 x 1000 + 5 x 100 + 4 x 10 + 6

44. 532235 = 5 x 100000 + 3 x 10000 + 2 x 1000 + 2 x 100 + 3 x 10 + 5

45. XXXIX=31

46. LXXIV=74

47. The number LIV is greater than LVI

48. The number 4578,4587,5478,5487 are in de scending order.

49. The number 85764 rounded off to nearest    hundreds is written as 85700.

50. Estimated sum of 7826 and 12469 rounded off to hundreds is 20,000

51. The largest six digit telephone number that can be formed by using digits 5,3,4,7,0,8 only once is 875403

52. The number 81652318 will be read as eighty one crore six lakh fifty two thousand three hundred eighteen

53. The largest 4-digit number formed by the digits 6,7,0,9 using each digit only once is 9760

54. Among kilo,milli and centi, the smallest is centi.

55. Successor of a one digit number is always a one digit numbers.

56. Successor of a 3-digit number is always a 3-digit number.

57. Predecessor of a two digit number is always a two digit number.

58. Every whole number has its successor.

59. Every whole number has its predecessor.

60. Between any two natural numbers, there is one natural number.

61. The smallest 4-digit number is tha successor of the largest 3-digit number.

62. Of the given two natural numbers, the one having more digits is greater.

63. Natural numbers are closed under addition.

64. Natural numbers are not closed under multiplication.

65. Natural numbers are closed under subtraction.

66. Addition is commutative for natural numbers.

67. 1 is the identity for addition of whole numbers.

68. 1 is the identity for multiplication of whole  numbers.

69. There is a whole number which when added to a whole number gives the number itself.

70. There is a natural number which when added to a natural number gives the number itself.

71. If a whole number is divided by another whole number, which is greater than the first one, the quotient is not equal to zero.

72. Any non-zero whole number divided by itself gives the quotient 1.

73. The product of two whole numbers need not be a whole number.

74. A Whole number divided by another whole number greater than 1 never gives the quo tient equal to the former.

75. Every multiple of a number is greater than or equal to the number.

76. The number of multiples of a given number is finite.

77. Every number is a multiple of itself.

78. Sum of two consecutive odd numbers is always divisible by 4.

79. If a number divides three numbers exactly , it must divide their sum exactly.

80. If a number exactly divides the sum of three numbers, it must exactly divide the numbers  separately.

81. If a number is divisible both by 2 and 3, then it is divisible by 12.

82. A number with three or more digit is divisible by 6, if the number formed by its last two digit (i.e.,ones and ten) is divisible by 6.

83. A number with 4 or more digit is divisible by 8, if the number formed by the last three digit is divisible by 8.

84. If the sum of the digit of a number is divisible by 3, then the number itself divisible by 9.

85. All numbers which are divisible by 4 may not be divisible by 8.

86. The Highest common factor of two or more numbers is greater than their lowest common multiple.

87. LCM of two or more numbers is divisible by their HCF.

88. LCM of two numbers is 28 and their HCF us 8.

89. LCM of two or more numbers may be one of the numbers.

90. HCF of two or more numbers may be one of the numbers.

91. Every whole numbers  is the successor of an other whole numbers.

92. Sum of two whole numbers is always less than their product.

93. If the sum of two distinct whole numbers is odd, then their difference also must be odd.

94. Any two consecutive numbers are coprime .

95. If the HCF of two numbers is one of the num- bers. them their LCM is the other number.

96. The HCF of two numbers is smaller then the smaller of the numbers

97. THe LCM of two numbers is greater than the larger of the numbers.

98. THe LCM of two coprime numbers is equal to the product of the numbers.

### OBJECTIVE :

1.D     2.A     3.B   4.C
5.B     6.C     7.D   8. D
9.D    10.B   11.C  12.A
13.B  14.C   15.B  16.C
17.C  18.B   19.D  20.C

### SUBJECTIVE :

1. Predecessor of 7000 = 7000 – 1 = 6999
2. 72389 < 72391
3. 2345, 2435, 2543, 3452, 4325
4. 62, 45, 635
9, 58, 61, 089
5. (a) 2,00,000 + 80,000 + 4,000 + 200 + 30 + 1
(b) 50,00,000 + 2,00,000 + 10,000 + 1,000 + 500 + 60 + 8
(c) 6,00,00,000 + 4,00,000 + 10,000 + 8,000 + 500 + 10 + 7
(d) 8,00,00,000 + 90,00,000 + 1,00,000 + 80,000 + 1,000 + 200 + 10 + 3
6. (a)

(b) one crore thirty lakh seventy nine
thousand six hundred ninty six
Thirty three crore seventy two lakh seventy two thousand one hundred fourten
Seventeen lakh sixty thousand six hundred seven.

(c) Thirten million seventy nine thousand six hundred ninty six.
Three hundred thirty seven million two
hundred seventy two thousand one hundred fourteen
One million seven hundred sixty thousand six hundred seven.

7. Greatest number : 97530
Smallest number : 30579
8.
9. (a) 3228 » 3200 ; 572 » 600
(b) 8010 » 8000 ; 2507 » 3000
(c) 32 » 30 ; 58 » 60
30 × 60 = 1800
(d) 108 × 47 ; 108 » 100
47 » 50
100 × 50 = 5000

10. (a) 46 = XLVI (b) 90 = XC (c) 120 = CXX (d) 150 = CL

11. (a) XX|| = 22 (b) C C X = 210 (c) D C C = 700 (d) L X l l l = 63

12. 7000 ml

13. 95,70,985

14. 9,00,085

15. 2,39,805

16. 1325 children

17. (i) 567, 576, 657, 675, 756, 765

(ii) 209, 290, 902, 920

18. (i) Greatest = 6521 Smallest = 1256
(ii) Greatest = 9870 Smallest = 7089
(iii) Greatest = 6543 Smallest = 3456
(iv) Greatest = 8432 Smallest = 2348
(v) Greatest = 9520 Smallest = 2059
(vi) Greatest = 9631 Smallest = 1369

19. (i) Greatest = 6632 Smallest = 2236
(ii) Greatest = 6610 Smallest = 1006
(iii) Greatest = 9974 Smallest = 4479
(iv) Greatest = 5520 Smallest = 2005

20. (i) Greatest = 6987 Smallest = 6012
(ii) Greatest = 9487 Smallest = 1402
(iii) Greatest = 9876 Smallest = 1072
(iv) Greatest = 9871 Smallest = 2031
(v) Greatest = 9876 Smallest = 9012
(vi) Greatest =9087 Smallest = 1023
(vii) Greatest = 9857 Smallest = 1052
(viii) Greatest = 9873 Smallest = 1023

### OBJECTIVE :

1. D 2. B 3. B 4. C
5. C 6. D 7. D 8. C
9. A 10. C 11. B 12. D
13. A 14. C 15. A 16. D
17. B 18. D 19. C 20. D
21. B 22. B 23. B 24. C
25. B 26. D

### SUBJECTIVE :

1. No, zero is the only number

2. Both travels equal distance (320 km)

3. Both sold equal number of tickets. (50 tickets)

5. (i) 2364   (ii) 3774

6. (i) Whole Number
(ii) sum, same, commutative property
(iii) 7 × (32 × 56) = (7 × 32) × 56
(iv) 0 (zero)

7. 10010

8. 147

9. (i) n = 5 (ii) n = 7 (iii) n = 8

10. (i) 9720 (ii) 0 (iii) 957000

11. Rs. 1160

12. (i) 0 (ii) 0 (iii) 700

13. (i) True (ii) False (iii) False (iv) True (v) False

14. 204

15. 19

16. 47

17. 2970

18. 11300

19. 444

20. Tuesday

21. (a) 507 (b) 20 (c) 400 (d) 11 (e) 974

22. (a) False (b) False (c) False (d) True (e) True

### OBJECTIVE :

1. C 2. B 3. A 4. B
5. B 6. D 7. C 8. C
9. B 10. A 11. C 12. D
13. C 14. A 15. D

### SUBJECTIVE :

1. 7
2. 1
3. 1
4. 0
5. 4
6. 1
7. (i) 1, 23
(ii) 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
(iii) 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168

8. (i) 112 (ii) 17, 51, 85
(iii) 55, 60, 65, 70, 75

9. 90, 91, 92, 93, 94, 95, 96
10. (i) 2 (ii) 2 (iii) 1
11. 4
12. 31L
13. 31
14. 20
15. (i) 300 (ii) 216 (iii) 5940
16. 151
17. 122m40cm
18. 9072
19. (a) 7, 11, 13, 17, 19, 23, 29, 31
(b) 71, 73, 79, 83, 89, 97
(c) 41, 43, 47, 53, 59, 61
(d) 79, 83, 89, 97, 101, 103, 109, 113, 127, 131, 139, 149, 151, 157
20. Yes, 9
21. (a) No (b) 1
(c) 4, 9, 25, 49
22. (a) 36 = 7 + 29 (b) 42 = 5 + 37 (c) 84 = 17 + 67 (d) 98 = 79 + 19

### CONCEPTUAL PROBLEMS FOR NUMBER SYSTEM

1. C    2. C     3. D     4. B    5. D    6. D     7. C    8. B    9. D    10.C    11.D    12.C    13.D    14.D    15.B    16.A    17.C    18.C    19.B    20.B    21.D    22.B    23.B    24.A

25.C    26.D   27.B    28.A    29.C   30.B    31.D   32.D    33.B    34.A    35.A    36.C    37.B    38.C    39.T    40.F     41.T    42.T    43.F    44.T     45.F     46.T

47. F   48. F  49. F   50. T    51. F   52. F    53. T  54. F    55. F   56. F   57. F    58. T    59. F    60. F   61. T   62. T   63. T   64. F    65. F   66. T    67. F    68. T

69. T  70. F   71. T   72. T    73. F   74. T    75. T  76. F   77. T    78. T   79. T   80. F    81. F    82. F    83. T  84. F   85. T   86. F     87. T   88. F   89. T    90. T

91. F   92. F   93. T   94. T   95. T   96. F   97. F   98. T