The gcd(a, b) = 18, lcm(a, b) = 108. if a=36, find b.

The way to solve this is by using Venn diagrams with one circle as a (36) and the other as b(?).  If a is 36, the factors of 36 are 3*3*2*2.  If 18 is the greatest common divisor, that goes in the overlap of a and b. So if the factors of 18 are 3*3*2, the common ones between that and 36 are 3*3*2 which leaves a 2 alone in the a circle. The least common multiple is found by multiplying all the factors together of a, b and the overlap.  If all the factors multiplied together equal 108, then 2(from the a side all alone)*(3*3*2)from the overlap*x(in the b circle)=108. Or in other words: 2*3*3*2*x=108 or 36x=108. If we solve for x we get that x=3. So b is the 18 in the overlap times 3, which is 54.