The concept of lcm (least common multiple)
Let n1 and n2 be two nautral numbers distinct from each other. The smallest natural number n that is exactly divisible by n1 and n2 is called the Least Common Multiple (LCM) of n1 and n2 is called the Least Common Multiple (LCM) of n1 and n2 and is designated as LCM (n1, n2).
Rule for Finding the LCM of 2 numbers n1 and n2
(a) Find the standard form of the numbers n1 & n2.
(b) write out all the prime factors, which are contained in the standard forms of either of the numbers.
(c) Raise each of the prime factors listed above to the highest of the powers in which it appears in the standard forms of the numbers n1 and n2.
(d) The product of results of the previous step will be the LCM of n1 and n2.
Ex. 1 Find the LCM of 150, 210, 375.
Step 1 : Writing down the standard form of numbers
150 = 5 × 5 × 3 × 2
210 = 5 × 2 × 7 × 3
375 = 5 × 5 × 5 × 3
Step 2 : Write down all the prime factors: That appear at least once in any of the numbers:
5, 3, 2, 7.
Step 3 : Raise each of the prime factors to their highest available power (considering each to the numbers).
ASSIGNMENT
1. Find the common factors for the numbers.
(A) 24 and 64 (B) 42, 294 and 882
(C) 60, 120 and 220
2. Find the HCF of
(A) 420 and 1782 (B) 36 and 48
(C) 54, 72, 198 (D) 62, 186 and 279
3. Find the LCM of
(A) 13, 23 and 48
(B) 24, 36, 44 and 62
(C) 22, 33, 45 and 72
(D) 13, 17, 21 and 33
4. Find the series of common multiples of
(A) 54 and 36 (B) 33, 45 and 60
[Hint: Find the LCM and then create an Arithmetic progression with the first term as the LCm and the common difference also as the LCM.]
5. The LCM of two numbers is 936. If their HCF is 4 and one of the numbers is 72, the other is:
(A) 42 (B) 52
(C) 62 (D) None of these
6. Two alarm clocks ring their alarms at regular intervals of 50 seconds and 48 seconds. If they first beep together at 12 noon, at what time will they beep again for the first time ?
(A) 12:10 P.M. (B) 12:12 P.M.
(C) 12:11 P.M. (D) None of these
7. 4 Bells toll together at 9>00 A.M. They toll after 7,8,11 and 12 seconds respectively. How many times will they toll together again in the next 3 hours ?
(A) 3 (B) 4 (C) 5 (D) 6
8. On Ashok Marg three consecutive traffic lights change after 36, 42 and 72 seconds respectively. If the lights are first switched one at 9:00 A.M. sharp, at what time will they change simultaneously ?
(A) 9 : 08 : 04 (B) 9 : 08 : 24
(C) 9 : 08 : 44 (D) None of these
10. A forester wants to plant 44 apple trees, 66 banana trees and 110 mango trees in equal rows (in terms of number of trees). Also, he want to make distinct rows of trees (i.e. only one type of tree in one row). The number of rows (minimum) that are required are :
(A) 2 (B) 3 (C) 10 (D) 11
11. Three runners running around a circular track, can complete one revolution in 2,4 and 5.5 hours respectively. When will they meet at the starting point ?
(A) 22 (B) 33 (C) 11 (D) 44
12. The HCF and LCM of two numbers are 33 and 264 respectively. When the first number is divided by 2, the quotient is 33. The other number is ?
(A) 66 (B) 132 (C) 198 (D) 99
13. The greatest number which will divide: 4003, 4126 and 4249.
(A) 43 (B) 41
(C) 45 (D) None of these
14. Which of the following represents the largest 4 digit number which can be added to 7249 in order to make the derived number divisible by each of 12, 14, 21, 33, and 54.
(A) 9123 (B) 9293
(C) 8272 (D) None of these
15. Find the greatest number of 5 digits, that will give us a remainder of 5, when divided by 8 and 9 respectively.
(A) 99931 (B) 99941
(C) 99725 (D) None of these
16. Find the greatest number of four digits which when divided by 10, 11, 15and 22 leaves 3, 4, 8 and 15 as remainders respectively.
(A) 9907 (B) 9903
(C) 9893 (D) None of these
17. What will be the least possible number of the planks, if three pieces of timber 42m, 49m and 63 m long have to be divided into planks of the same length ?
(A) 7 (B) 8
(C) 22 (D) None of these
18. Find the greatest number, which will divide 215, 167 and 135 so as to leave the same remainder in each case.
(A) 64 (B) 32 (C) 24 (D) 16
19. Find the L.C.M of 2.5, 0.5 and 0.175
(A) 2.5 (B) 5 (C) 7.5 (D) 17.5
20. The L.C.M of 4.5; 0.009; and 0.18 = ?
(A) 4.5 (B) 45 (C) 0.225 (D) 2.25
21. The L.C.M of two numbers is 1890 and their H.C.F is 30. If one of the m is 270, the other will be
(A) 210 (B) 220 (C) 310 (D) 320
22. What is the smallest number which when increased by 6 is divisible by 36, 63 and 108?
(A) 750 (B) 752 (C) 754 (D) 756
23. The smallest square number, which is exactly divisible by 2, 3, 4, – 9, 18, 36 and 60, is
(A) 900 (B) 1600
(C) 3600 (D) None of these
24. The H.C.F of two numbers is 11, and their L.C.M is 616. If one of the numbers is 88, find the other ?
(A) 77 (B) 87
(C) 97 (D) None of these
25. The HCf and LCm of two numbers are 12 and 144 respectively. If one of the number sis 36, the other number is
(A) 4 (B) 48 (C) 72 (D) 432
ANSWER KEY
5) B 6) D 7) C 8) B 9) C 10) C 11) D 12) B 13) B 14) B 15) B 16) C 17) C 18) D 19) D 20) A
21) A 22) A 23) A 24) A 25) B