tofubeans 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

The answer is 12, but I don't really understand the explanation the book gives.

The question stem says, n^2 is divisible by 72.
i.e n^2 = 72*k where k is an integer.
n^2 = 36*2*k = (6^2)*2*k.

Since (6^2)*2*k (= n^2) is a perfect square, 2*k should be a perfect square(Since 6^2 is already a perfect square). For 2*k to be a perfect square, the least value of k is 2(Since k is an integer) and the integer k should be of the form 2*L, where L is a square of an integer greater than 0.

Now, we know that n^2 = 72*k = 72*2*L = 144*L, where L is a square of an integer greater than 0. Applying sqaure root on both the sides, we get
n = 12*(a positive integer = √L). So the largest positive integer that must divide n is 12.

n^2 = 72I, where I is an integer
n^2 = 2*2*2*3*3*I

Since n^2 is square of n, what is the smallest number it should be a multiple of so that n^2 can be represented by square of a number?
2*2*3*3 is a possibility, but it's too big. Is there anything smaller?
3*3 is a possibility, but it's too big. Anything smaller?
The smallest number is 2, and let I = 2 J^2, where J is an integer. Then

n^2 = (2*2*2*2*3*3)*(J^2) = (2*2*3*J)^2
or
n = 2*2*3*J, or a multiple of 12.

(B) 12 is the answer.

Hey Anurag,

In the picking number approach, i picked 144 - but the largest number that divides 144 is 48 (one of the choices). I am unable to understand why it is not the answer. Could you please explain?

Thanks!

Anurag@Gurome wrote:

tofubeans 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

The answer is 12, but I don't really understand the explanation the book gives.

Picking Number Approach

• Least possible value of n² such that n² is divisible by 72 is 72*2 = 144
  Hence, minimum possible value of n = 12.
  Largest possible integer that divides n is 12.

Algebraic Approach:

• n² is divisible by 72
  Hence we can write n² as 72k, where k is an positive integer.
  Now, n = √n² = √(72k) = √[(2)*(36)*k] = 6√(2k)
  Now for n to be an integer, k must be an even multiple of a perfect square.
  Hence, we can write k = 2m², where is a positive integer.
  Now, n = 6√(2k) = 6√(2*2*m²) = 12m
  
  Hence, largest possible integer that divides n is 12

The correct answer is B.

Thank you. i was over thinking it and taking n as 144!

terrible mistake! anyway, got it now. Thank you very much.

Anurag@Gurome wrote:

hulahooporiginal wrote:Hey Anurag,

In the picking number approach, i picked 144 - but the largest number that divides 144 is 48 (one of the choices). I am unable to understand why it is not the answer. Could you please explain?

Thanks!

You picked 144 as the value of n², right? This means that n = 12. But the largest possible integer that divides n is 12. 48 is not possible. I hope you get the point?