nikhilagrawal wrote:If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be
A. 2
B. 5
C. 6
D. 7
E. 14
Some good solutions, let's review the general principle.
All numbers are made up of prime factors. All perfect squares are made up of pairs of primes.
So, in order for 3150y to be a perfect square, it must consist only of pairs of prime factors.
There are many different ways to find the prime factors of a number - my personal preference is a prime factorization tree (which is a bit hard to show on the computer).
3150 breaks up into 315 * 10 (no primes)
315 breaks up into 5 * 63 (5 is prime, put that aside)
63 breaks up into 7*9 (7 is prime, put that aside)
9 breaks up into 3*3 (both prime, put them aside).
10 breaks up into 2*5 (both prime, put them aside).
So, we've put aside: 5*7*3*3*2*5
writing in order:
2*3*3*5*5*7
Our 3s and 5s are paired off, so we don't need any more of those to create a perfect square.
We have a single 2, so we need a 2.
We have a single 7, so we need a 7.
So, the minimum possible value for y is 2*7 = 14... choose E.
