We don't need exact values for n and y (it's impossible to find exact values), but we do need to calculate the minimum value for y.doudou wrote:If n and y are positive integers and 450y = n^3, which of the following must be an integer?
1 - y/(3*2^2*5)
2 - y/(3^2*2*5)
3 - y/(3*2*5^2)
To do so, start by isolating y:
y = n^3/450
Next, let's break 450 down into it's primes:
450 = 5 * 90 = 5 * 5 * 18 = 5 * 5 * 2 * 9 = 5 * 5 * 2 * 3 * 3
Now, since we know that n is an integer, n^3 must have triplets of all of it's primes (e.g. if n=5, then n^3 = 5*5*5).
If n^3/450 is an integer, then all of the prime factors of 450 must be prime factors of n^3 as well.
Therefore, 5, 5, 2, 3 and 3 are prime factors of n^3. Since all primes of n^3 will appear in triplets, to round out the value of n^3 we're going to need, at a minimum, one more 5, two more 2s and one more 3.
So, the minimum value of N^3/450 = 5*2*2*3.
Since this minimum value is also the minimum value of y (since n^3/450=y), all values of y must be divisible by 5*2*2*3, which is answer choice (1).
