Problem Solving

LUANDATO wrote:If n is a positive integer, and n^2 has 25 factors, which of the following must be true.

1. n has 12 factors.
2. n > 50
3. √n is an integer.

A. 1 and 2
B. 2 only
C. 3 only
D. 2 and 3
E. none

The OA is C.

Can any expert help me with this PS question please? I don't understand it. Thanks.

These are certain basic formulas pertaining to factors of a number N, such that,

N = p^a.q^b.r^c; where, p, q and r are prime factors of the number n and a, b and c are nonnegative powers/ exponents.

Number of factors of N = (a +1).(b +1).(c +1)

Coming to the question.

We have n^2 has 25 factors.

Say n^2 when prime-factorized has a, b and c as nonnegative powers/ exponents.

Since 25 factored as 5 x 5, each 5 would have been formed by (4 + 1), thus n^2 when prime-factorized has 4 and 4 as nonnegative powers/ exponents.

Thus, n^2 = p^4.q^4; where, p, q and r are prime factors of n^2

Thus, n = √(n^2) = √(p^4.q^4); where, p, q and r are prime factors of n

=> n = p^2.q^2

Let's see each statement one by one.

1. n has 12 factors: The number of factors of n = (2 + 1).(2 + 1) = 9 factors. This is incorrect.
2. n > 50: We have n = p^2.q^2, but we do not know the values of p and q. If p = 2 and q = 3, then n = 36. This is incorrect.
3. √n is an integer: We have n = p^2.q^2, thus √n = √(p^2.q^2) = pq = Integer. Correct.

The correct answer: C

Hope this helps!

-Jay

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