If the LCM of a amd 12 is 36 what could be the possible values of a?

Here is what I do in my head (mental tricks):

LCM (12,36) --> (12/12,36/12) --> (1,3) --> 3*12 or 1*36 --> 36
LCM (12, 30) --> (12/6,30/6) --> (2,5) --> 5*12 or 2*30 --> 60
LCM (10, 28) --> (10/2,28/2) --> (5,14) --> 14*10 or 5*28 --> 140

GCD (12,36) --> (12/12,36/12) --> (1,3) --> 12
GCD (12, 30) --> (12/6,30/6) --> (2,5) --> 6
GCD (10, 28) --> (10/2,28/2) --> (5,14) --> 2

LCM and GCD have a nice property: LCM(x,y)*GCD(x,y) = xy

For example, LCM(12,30)*GCD(12,30) = 60*6 = 12 * 30

i thank both of you, but i did not get walker's approach.can you please sharpen your explenation?

let's say that there are 3 numbers:
440,120 and 80.how can you use your shortcut

LCM (80, 120, 440) --> (80/40, 120/40, 440/40) --> (2, 3, 11) --> 3*11*80 --> 2640

GCD (80, 120, 440) --> (80/40, 120/40, 440/40) --> (2, 3, 11) --> 40

40 here is the greatest common divisor.

12 = (2^2)*3
a = (2^2 or 2^1 or 2^0)*(3^2)

L.C.M of a & 12 = 36 = (2^2 )*(3^2) = product of Maximum powers of each prime factor

This can be easily understood by comparing 12 with 36 .

12 has one 3 , but L.C.M which is the maximum powers of each prime factor has two 3's

=> a must have two 3's in it = 3^2 ------1

12 has two 2's. L.c.m which is the maximum powers of each prime factor has two 2's as well.

=> a could have zero 2's or one 2 or two 2's------2

clubbing above two statements marked 1 *2 , we have a = (2^2 or 2^1 or 2^0)*(3^2)
=(4 or 2 or 1)*9
=> a could be 36 or 18 or 9.

Hope its clear now.

Thanks Karishma. Your solution is very clear and in simple way. I didn't understand the complicated solution prepared by Manhattan. Their prime columns technique is good but solution explanation was confusing.

KarishmaB

Why can't a potential value for a be just 3?
My reasoning is 3*12=36.

Many thanks!

Because LCM is the lowest common multiple. It is the product of the two numbers only when they have no common factors.
LCM of 3 and 12 will not be 36. It will be 12.

A = x*y*z*....
12 = 2^2*3

LCM(A,12) = 36 = 2^2*3^2

The least common of A and 12 must have highest power factors of each of them that constitute LCM 36
2^2 is with 12 so 3^2 must be with A.
9 <= A = 9*multiples

However, what multiples we can have. We can have only 2 with highest power of 2.
Thus, A can be
9*1 = 9
9*2 = 18
9*2^2 = 36

Any more multiples would not result in LCM of 36.

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