Problem Solving

Set X, Y, and Z consists positive multiples of 4, 5, and 6, respectively. If Set A is the
intersection of X and Y, set B is the intersection of Y and Z, what is the least possible number
of intersection of A and B?
A. 30
B. 60
C. 90
D. 120
E. 240


B

apoorva.srivastva wrote:Set X, Y, and Z consists positive multiples of 4, 5, and 6, respectively. If Set A is the
intersection of X and Y, set B is the intersection of Y and Z, what is the least possible number
of intersection of A and B?
A. 30
B. 60
C. 90
D. 120
E. 240


B

Obviously 60.

A the least X int Y is 20 and B the least Y int Z is 30; therefore the least possible number of intersection of A and B is the LCM of 20 and 30 that is 60.

Set A = Intersection of X & Y => Set A contains multipes of 20
Set B = Intersection of Y & Z => Set B contains multiples of 30

Least possible number of Intersection of Set A & B is nothing but LCM of 20 & 30, which is 60.
Hence ans is B.

what is LCM? I got the 20 and 30 but I'm clueless on the whole picture of this problem.

diegow77 wrote:what is LCM? I got the 20 and 30 but I'm clueless on the whole picture of this problem.

This question is a very convoluted means of asking:

"What's the lowest common multiple of 4, 5 and 6?"

("LCM" = "Lowest Common Multiple".)

Set A is all the common multiples of 4 and 5; set B is all the common multiples of 5 and 6. So, the smallest member common to both sets A and B will be the lowest common multiple of 4, 5 and 6, which is 60.

To find the lowest common multiple, we can either use brute force or prime factors.

Brute force is pretty quick when dealing with small numbers. We just write all the multiple of the biggest number until we find a number that's divisible by all the numbers on our list:

6, 12, 18, 24, 30, 36, 42, 48, 54, 60... done!

Of course, we can use common sense/logic to speed up the process; we know we need a "5" in our number, so we can just jump to multiples of both 5 and 6:

30 (4 doesn't go in)
60 (4 does go in - done!)

We can also use primes:

4 = 2*2
5 = 5
6 = 2*3

So, our LCM must contain two "2"s, one "3" and one "5" to accomodate all our numbers:

2*2*3*5 = 60

Thanks a lot Stuart, you're a genius!