8
Least common multiple
Textbook, pp. 26–29
GoalFind the least common multiple (LCM) and see how it relates to the GCD.
New words
common multiple · umumiy karralileast common multiple (LCM) · EKUKsmallest · eng kichikproduct · ko‘paytma
Explanation
A number that is a multiple of both of two numbers is a common multiple of them. The smallest common multiple is the least common multiple (LCM). To find it, factor the numbers into primes and multiply all prime factors that occur, each with its greatest power. For example 12 = 2² · 3 and 18 = 2 · 3², so LCM = 2² · 3² = 36. For coprime numbers the LCM is their product. The product of two numbers equals the product of their GCD and LCM.
Worked examples
LCM(4, 6): multiples of 4 are 4, 8, 12, ...; multiples of 6 are 6, 12, ... The first common one is 12.
Two buses leave a stop every 12 and 18 minutes. After leaving together they next leave together 36 minutes later, since LCM(12, 18) = 36.
Class activity
“Two lamps”: two lamps are drawn on the board; one blinks every 6 seconds, the other every 8. Pupils mark the blinks on a time line and find the first moment both blink together.
Practice
1
Find LCM(6, 8).
24
2
5 and 7 are coprime. Find their LCM.
35
3
Two lamps blink every 15 and every 20 seconds. After blinking together, after how many seconds do they blink together again?
60
4
GCD(12, 18) = 6. Use this to find LCM(12, 18).
36