If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is
A) 6
B) 12
C) 24
D) 36
E) 48
A) 6
B) 12
C) 24
D) 36
E) 48
The key word here is MUST.runningguy wrote:If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is
A) 6
B) 12
C) 24
D) 36
E) 48
In order to find the largest positive integer that must divide n, we need to find the smallest possible value of n.
The question tells us that n² is divisible by 72.
So, n² is a multiple of 72 AND n² is the square of an integer.
Let's list some squares of integers: 1, 4, 9, 16, . . . 100, 121, 144, 169...
So, 144 is the smallest perfect square that is divisible by 72
If n² = 144, then n = 12.
So, 12 is the smallest possible value of n that satisfies the given conditions.
If n = 12, then the largest positive integer that must divide n is 12 = B
Aside: Notice that 12 is NOT DIVISIBLE by 24, 36 or 48 (so we can eliminate C, D and E). This shows why we wanted to find the smallest possible value of n.
Cheers,
Brent
