Properties of perfect squares

[ayushiiitm]

If the positive integer N is a perfect square, which of the following must be true?

I. The number of distinct factors of N is odd.
II. The sum of the distinct factors of N is odd.
III. The number of distinct prime factors of N is even.

Options are
I only
II only
I and II
I and III
I, II and III

My take:
N is a perfect square
statement 1 seems to satisfy
25=5, 36=6 (so for perfect square the factors are paired up, hence number of factors is odd)

Statement2and statement3 :
How to solve this? any property?

or is it by putting values and checking

[tpr-becky]

Your explanation for statement I does not consider all of the factors of the number- it only says distinct factors, not distinct prime factors

therefore if N=25 then the distinct factors are 1, 5, and 25 - and odd number and if N=36 then the distinct factors are 1, 2, 3, 4, 6, 9, 12, 18 and 36 - another odd number - same with 4 - therefore I believe sTatement #1 must be true.

Statement #2 - you can use teh above examples - 1+5+25 = odd, 1+2+3_4_6_9_12+18+36 is odd. - this therefore statement #2 seems to satisfy.

Statemtn 3 now talks about distinct prime factors - for 25 it is 1 prime factor(5), for 36 there is 2 and 3 - so this does not work.

is C the correct choice.

[ayushiiitm]

So we can solve it by putting in values...right?

No number property

neways Thanks Becky

[tpr-becky]

Because this problem is so specific I would solve it by plugging in numbers - but since Stmt 1 and 2 always work then those are number properties. It just wouldn't be efficient to try to memorize all of the number problems for this exam.

[venmic]

I have read in a post and on wiki that 1 need not be considered it can go in everything so not

So if N = 25 then the factors are 5,25 = so the number of factors are even in case of a perfect square

Sum of the factors is even too = 30

PLEASE explain

Your explanation for statement I does not consider all of the factors of the number- it only says distinct factors, not distinct prime factors

therefore if N=25 then the distinct factors are 1, 5, and 25 - and odd number and if N=36 then the distinct factors are 1, 2, 3, 4, 6, 9, 12, 18 and 36 - another odd number - same with 4 - therefore I believe sTatement #1 must be true.

Statement #2 - you can use teh above examples - 1+5+25 = odd, 1+2+3_4_6_9_12+18+36 is odd. - this therefore statement #2 seems to satisfy.

Statemtn 3 now talks about distinct prime factors - for 25 it is 1 prime factor(5), for 36 there is 2 and 3 - so this does not work.

is C the correct choice.

[Ian Stewart]

I have read in a post and on wiki that 1 need not be considered it can go in everything so not

So if N = 25 then the factors are 5,25 = so the number of factors are even in case of a perfect square

Sum of the factors is even too = 30

PLEASE explain

You certainly need to consider 1 when you count the divisors of a positive integer, so you may have gotten bad information from what you read. Perfect squares have an odd number of divisors. It's easy to see why this should be true. If you take a non-square, like 18, then all of the divisors are in pairs:

1, 18
2, 9
3, 6

If you take a perfect square, however, one divisor will not be in a pair (the square root of the number). For example, the divisors of 36 are

1, 36
2, 18
3, 12
4, 9
6

So perfect squares have an odd number of divisors, and all other numbers have an even number of divisors.

I've solved the rest of the question in the original post a few times elsewhere, but I'd point out that item II in the list (about the sum of a square's divisors) is irrelevant for the GMAT. It's not a property you'd ever want to memorize for the test, since it will never be tested, nor is it the kind of thing you could reasonably be asked to prove is true within two minutes. GMAT questions are never designed so that the only viable strategy is making a guess by picking a few numbers.

[Scott@TargetTestPrep]

If the positive integer N is a perfect square, which of the following must be true?

I. The number of distinct factors of N is odd.
II. The sum of the distinct factors of N is odd.
III. The number of distinct prime factors of N is even.

Options are
I only
II only
I and II
I and III
I, II and III

Solution:

Recall that the number of factors of a perfect square is odd (e.g., 1 has 1 factor, 4 has 3 factors, 9 has 3 factors, 16 has 5 factors, and so on). So statement I is true.

Another fact about the distinct factors of a perfect square is that their sum is odd. (e.g., the distinct factors of 4 are 1, 2, and 4, which sum to 7; the distinct factors of 9 are 1, 3, and 9, which sum to 13, and so on.) Thus, statement II is true.

Since 4 = 2^2 only has 1 distinct prime factor (namely 2), Statement III is not true.

Answer: C

[Neha sharma]

The perfect square number has following properties

1. The number of distinct factors of a perfect square is ALWAYS ODD.
2. The sum of distinct factors of a perfect square is ALWAYS ODD
3. Perfect square always has even powers of its prime factors.

So 1 & 2 statement must be true
Imo-C