2013 AMC 10B Problem 9

Problem 9 of 25EasierNumber Theory

Three positive integers are each greater than

1,

have a product of

27000 ,

and are pairwise relatively prime. What is their sum?

Show solution

Solution

Only one number is a multiple of

2,

only one is a multiple of

3,

and only one is a multiple of

5.

Since each positive integer is greater than

1,

each must be a multiple of one of these primes. Now

27000=2^3\cdot 3^3\cdot 5^3.

Therefore, the numbers must be

2^3,3^3,5^3,

and their sum is

160.

Thus, the correct answer is D.

Tagged: prime factorization · greatest common divisor