UNO High School Problem of the Week Competition
Problem 15
The positive integers 30, 72, and N are related to each other such that the product of any two of them is divisible by the third. What is the smallest possible value of N?
Solution
First, observe that 30 divides 72N. This means that there are factors of 2, 3, and 5 in 72N. Note 2 and 3 are factors of 72, while 5 is not. Thus 5 must divide N. Secondly, 30N is a multiple of 72, so N must be a multiple of 12. The smallest positive integer with 5 and 12 as factors is 60. To confirm that 60 is indeed the smallest valid value of N, we note that 60 divides into 30 * 72 and therefore satisfies all of the requirements.
