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Is the positive integer N a perfect square? (1) The number 17 Feb 2015, 08:58
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Hi @veritasprepkarishma/@Bunuel,

For 2nd statement if we take
1)4-perfect square-sum of distinct factors is 2 or 4(2*2 or 4*1)
Condition satisfied

2)8-not a perfect square-sum of distinct factors is 2 or 8(2*2*2 or 8*1)
Condition satisfied still not perfect square


Then how can D be the answer?
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Is the positive integer N a perfect square? (1) The number 17 Feb 2015, 09:39
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ssriva2
Hi @veritasprepkarishma/@Bunuel,

For 2nd statement if we take
1)4-perfect square-sum of distinct factors is 2 or 4(2*2 or 4*1)
Condition satisfied

2)8-not a perfect square-sum of distinct factors is 2 or 8(2*2*2 or 8*1)
Condition satisfied still not perfect square


Then how can D be the answer?
Show more

(2) says that the sum of all distinct factors of N is even.

If N = 4, then its factors are 1, 2, and 4 --> the sum = 1 + 2 + 4 = 7 = odd.

If N = 8, then its factors are 1, 2, 4 and 8 --> the sum = 1 + 2 + 4 +8 = 15 = even.
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Is the positive integer N a perfect square? (1) The number 18 Feb 2015, 11:21
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goldeneagle94
Interesting Question !!!

A few facts to review:


A perfect sqaure ALWAYS has an ODD number of factors, whose sum is ALWAYS ODD.

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

Using the above facts, you can conclude that both statements are sufficient to answer the question.
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Perfect Square 36:
(6 x 6)
(3 X 3 X 2 X 2)
4 total factors, 2 distinct factors, and sum is even…?
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Is the positive integer N a perfect square? (1) The number 18 Feb 2015, 21:33
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Hi Kitzrow,

You have to note the difference between "factors" and "prime factors"

36 has the following FACTORS:
1 and 36
2 and 18
3 and 12
4 and 9
6

So, there are 9 factors and the sum of those factors is 91.

This example matches the prior statements - 36 has an ODD number of factors and the sum of those factors is ODD.

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Is the positive integer N a perfect square? (1) The number 05 Dec 2019, 23:40
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A perfect square represents the square of an integer.

Questions on the properties of a perfect square are quite common on the GMAT. So, it’s important that you devote enough time to learn the unique characteristics of perfect squares, so that you will be able to deal with these questions. A good way to start would be to memorise the first 25 perfect squares and observe patterns.

Two important properties that we will need to use to solve this question are:

1) A perfect square always has an odd number of distinct factors. The vice-versa is true as well; any number that has an odd number of distinct factors has to be a perfect square.
For example, the number 4 has 3 factors viz., 1,2 and 4; similarly, the number 16 has 5 factors viz. 1, 2, 4, 8 and 16.

2) A perfect square always has an odd number of odd factors and an even number of even factors. This means that the sum of the factors of a perfect square is going to be odd.

Taking the same numbers, 4 and 16, we see that the sum of the factors of 4 is 7 and the sum of the factors of 16 is 31.

If there are odd number of odd factors, the sum of these will be odd; even number of even factors will add up to an even number. Adding an odd sum with an even sum will give us the final sum as odd. That is how the sum of the factors of a perfect square is always odd.

Let us analyse the statements now.

Statement I alone says that the number of distinct factors of N is even. This only means that N is NOT a perfect square. We can answer the question with a NO. Remember that a NO is as good an answer as a YES, in a Yes-No type of DS question.

Statement I alone is sufficient. Possible answer options are A or D. Answer options B, C and E can be eliminated.

Statement II alone says that the sum of all distinct factors is even. Again, this means that N is NOT a perfect square and hence we can answer the question with a NO.
Statement II alone is sufficient. Answer option A can be eliminated.
The correct answer option is D.

Hope that helps!
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Is the positive integer N a perfect square? (1) The number 24 Aug 2023, 08:22
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Bunuel
Tips about the perfect square:
1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);

4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: \(36=2^2*3^2\), powers of prime factors 2 and 3 are even.

NEXT:
There is a formula for Finding the Number of Factors of an Integer:

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.

Back to the original question:

Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even --> let's say \(n=a^p*b^q*c^r\), given that the number of factors of \(n\) is even --> \((p+1)(q+1)(r+1)=even\). But as we concluded if \(n\) is a perfect square then powers of its primes \(p\), \(q\), and \(r\) must be even, and in this case number of factors would be \((p+1)(q+1)(r+1)=(even+1)(even+1)(even+1)=odd*odd*odd=odd\neq{even}\). Hence \(n\) cannot be a perfect square. Sufficient.

(2) The sum of all distinct factors of N is even --> if \(n\) is a perfect square then (according to 3) sum of odd factors would be odd and sum of even factors would be even, so sum of all factors of perfect square would be \(odd+even=odd\neq{even}\). Hence \(n\) cannot be a perfect square. Sufficient.

Answer: D.

Hope it helps.
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mbaMission
Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even.
(2) The sum of all distinct factors of N is even.
Show more


hi, i had a doubt with respect to statement 1:
if we take the example of a prime number, say 7, its factors are 7 and 1
therefore the number of distinct factors is 2 ie even

in that case, how would statement 1 be sufficient to prove that N is a perfect square?
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Is the positive integer N a perfect square? (1) The number 24 Aug 2023, 08:37
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Bunuel
Tips about the perfect square:
1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);

4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: \(36=2^2*3^2\), powers of prime factors 2 and 3 are even.

NEXT:
There is a formula for Finding the Number of Factors of an Integer:

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.

Back to the original question:

Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even --> let's say \(n=a^p*b^q*c^r\), given that the number of factors of \(n\) is even --> \((p+1)(q+1)(r+1)=even\). But as we concluded if \(n\) is a perfect square then powers of its primes \(p\), \(q\), and \(r\) must be even, and in this case number of factors would be \((p+1)(q+1)(r+1)=(even+1)(even+1)(even+1)=odd*odd*odd=odd\neq{even}\). Hence \(n\) cannot be a perfect square. Sufficient.

(2) The sum of all distinct factors of N is even --> if \(n\) is a perfect square then (according to 3) sum of odd factors would be odd and sum of even factors would be even, so sum of all factors of perfect square would be \(odd+even=odd\neq{even}\). Hence \(n\) cannot be a perfect square. Sufficient.

Answer: D.

Hope it helps.

mbaMission
Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even.
(2) The sum of all distinct factors of N is even.


hi, i had a doubt with respect to statement 1:
if we take the example of a prime number, say 7, its factors are 7 and 1
therefore the number of distinct factors is 2 ie even

in that case, how would statement 1 be sufficient to prove that N is a perfect square?
Show more

From (1) we deduce that n cannot be a perfect square, so the answer to the question is NO. Your example, also gives a NO answer to the question, so it aligns with the deduction.
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Is the positive integer N a perfect square? (1) The number 24 Aug 2023, 21:50
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TA396
Bunuel
Tips about the perfect square:
1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);

4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: \(36=2^2*3^2\), powers of prime factors 2 and 3 are even.

NEXT:
There is a formula for Finding the Number of Factors of an Integer:

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.

Back to the original question:

Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even --> let's say \(n=a^p*b^q*c^r\), given that the number of factors of \(n\) is even --> \((p+1)(q+1)(r+1)=even\). But as we concluded if \(n\) is a perfect square then powers of its primes \(p\), \(q\), and \(r\) must be even, and in this case number of factors would be \((p+1)(q+1)(r+1)=(even+1)(even+1)(even+1)=odd*odd*odd=odd\neq{even}\). Hence \(n\) cannot be a perfect square. Sufficient.

(2) The sum of all distinct factors of N is even --> if \(n\) is a perfect square then (according to 3) sum of odd factors would be odd and sum of even factors would be even, so sum of all factors of perfect square would be \(odd+even=odd\neq{even}\). Hence \(n\) cannot be a perfect square. Sufficient.

Answer: D.

Hope it helps.

mbaMission
Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even.
(2) The sum of all distinct factors of N is even.


hi, i had a doubt with respect to statement 1:
if we take the example of a prime number, say 7, its factors are 7 and 1
therefore the number of distinct factors is 2 ie even

in that case, how would statement 1 be sufficient to prove that N is a perfect square?
Show more

Also check out the following resources on Factors:

Blog Posts:
https://anaprep.com/number-properties-f ... -a-number/
https://anaprep.com/number-properties-r ... e-factors/

Video:
https://youtu.be/DxIH8rjhpKY
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Is the positive integer N a perfect square? (1) The number 11 Sep 2025, 08:56
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