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. A) I only B) II only C) I and II D) I and III E) I, II and III For III, 1 is not considered as prime factor. So, for example, for 4, distinct prime factor would be 2 only and not 2 and 1 both. Likewise, for 9, distinct prime factor would be 3 only Please write if my understanding is correct.

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If the positive integer N is a perfect square, which of the following

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Bunuel

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Bunuel

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.

Yes, your understanding of III is right. Prime factor of 4 is 2 and prime factor of 9 is 3. So III is not alway true: a perfect square can have any number of prime factors.

Tips about the perfect square:

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. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.
4. Perfect square always has even powers of its prime factors.

So I and II must be true.

what about for (iii) if N = 1? then 1^2 = 1
therefore 0 even

N = 2 then 2^2 = 4
1, 2, 4
2 even factors

N = 3 then 3^2 = 9
1, 3, 9
0 even factors?

Yes thats what I mean...then isnt 3 true? why is only I and II true?

The question asks: which of the following must be true?

III says: the number of distinct prime factors of N is even.

III is not always true: a perfect square can have any number of prime factors. For example, 4 has only one prime factor, which is 2.

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02 May 2017, 05:56

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Is that the case of GMAT? Well in general Mathematics, I'm sure factors can be either positive or negative. It's basically any number that divides a given number.

Bunuel

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02 May 2017, 07:07

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sarathgopinath92

Is that the case of GMAT? Well in general Mathematics, I'm sure factors can be either positive or negative. It's basically any number that divides a given number.

Factor is a positive divisor (at least on the GMAT).

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Bunuel

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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.

For III, 1 is not considered as prime factor.
So, for example, for 4, distinct prime factor would be 2 only and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 only

Please write if my understanding is correct.

Yes, your understanding of III is right. Prime factor of 4 is 2 and prime factor of 9 is 3. So III is not alway true: a perfect square can have any number of prime factors.

Tips about the perfect square:

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. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.
4. Perfect square always has even powers of its prime factors.

So I and II must be true.

Let me try to explain it algebraically.
The perfect number A is factored as : a^2 x b^2 x c^2 (the power is always EVEN, for the sake of simplicity, I use the power of 2).
1. The # of distinct factors = 3 x 3 x 3 = 27 : ODD
2. The sum of distinct factors = (a^3 - 1) (b^3 - 1) (c^3 - 1) / [ (a-1)(b-1)(c-1) = (a^2 + a + 1)(b^2 + b + 1)(c^2 + c + 1)
because a,b,c are different primes then there is at most one even factor among a, b, and c. Let's say a = 2 -> (a^2 + a + 1) = EVEN + EVEN + ODD = ODD
b, c must be ODD -> (b^2 + b + 1) = ODD + ODD + ODD = ODD and (c^2 + c + 1) = ODD + ODD + ODD = ODD.
SO (a^2 + a + 1)(b^2 + b + 1)(c^2 + c + 1) = ODD x ODD x ODD = ODD.
3. A = a^2 x b^2 x c^2 if a is EVEN and b, c are ODD.
The # of possible factors of A = 3 x 3 x 3 = 27 = ODD.
The # of possible factors not containing a (so will be ODD) = 3 x 3 = 9 = ODD.
-> The # of possible factors containing a (so will be EVEN) is : 27 - 9 = 18 = ODD - ODD = EVEN.

Hope it helps.

A small addition to the above. Even if a is not even, (a^2 + a + 1) will result be odd. i.e. (ODD+ODD+ODD)=ODD

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25 May 2020, 01:23

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Bunuel

Orange08

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.

For III, 1 is not considered as prime factor.
So, for example, for 4, distinct prime factor would be 2 only and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 only

Please write if my understanding is correct.

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.

So, in case of 9, factors are 3 and 9, so both 3 and 9 (distinct factors) are odd but 2 (even) in number. Please help me understand.

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25 May 2020, 01:29

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aditiphadnis

Bunuel

Orange08

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.

For III, 1 is not considered as prime factor.
So, for example, for 4, distinct prime factor would be 2 only and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 only

Please write if my understanding is correct.

You are missing that a number is a factor of itself, so factors of 9 are 1, 3, and 9.

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Orange08

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

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For statement 3 what happens if we consider 36 (6^2).

Prime factorisation : 2^2*3^2
Total factors (2+1)(2+1)=9
Distinct prime factors: 2 & 3

Therefore won't statement 3 be true in this case since it has 2 distinct prime factors?

Please correct my understanding.

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23 Jan 2024, 01:53

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MishalSen1

Isn't that doubt addressed HERE. Have you missed that post?

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