Problem Solving

If 5,400n is the square of an integer, what is the smallest possible integer value of n?
(A) 2
(B) 3
(C) 5
(D) 6
(E) 15

How should I read 5,400n? I know that I should use prime factorization to solve this problem. My struggle is...should I apply the logic to 5400 or 5400(2/3/5/6/15)? Any pointers would be much appreciated.

54*100=6*9*4*25=3^3*2^3*5^2


3^3*2^3*5^2*n=(some integer)^2

so, n =3*2=6

Pugalenthi wrote:If 5,400n is the square of an integer, what is the smallest possible integer value of n?
(A) 2
(B) 3
(C) 5
(D) 6
(E) 15

How should I read 5,400n? I know that I should use prime factorization to solve this problem. My struggle is...should I apply the logic to 5400 or 5400(2/3/5/6/15)? Any pointers would be much appreciated.

The 1st thing you need to remember is that a perfect square contains only even powers. Start by factorising the number given:
=54(100)
=9x6x10x10
=3x3x3x2x2x5x2x5
2^3 x 3^3x 5^2

Notice that the powers of 2 and 3 are odd implying that 5400 is not a square. 5400 x n must yield in a prime factorisation with all even powers. This therefore means that n must have at least one 2 and one 3 to make it: 2^4 x 3^4 x 5^2. Hence n can only be 2 x 3 = 6 ----choose D

Thank you for your reply and I understand the logic but my question is this....5400n ..given this number...why split this as 54*100 ? Why it shouldn't be 54002?

Your way of thinking is different. I give you that.

But do tell me, do you consider 2x as 2*x or something like, 22, 23, 24....29 ?

Get my point?