Is the positive integer N a perfect square? (1) The number

Question

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.

Bunuel · Accepted Answer

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
eq{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
eq{even} . Hence  n  cannot be a perfect square. Sufficient.

Answer: D.

Hope it helps.

Bunuel · Answer

2. Properties of Integers  
     Theory   
 Math: Number Theory 
 Number Properties: Tips and hints 
 Advanced Number Properties on GMAT (All 5 parts) 
 Do you make these 3 mistakes in GMAT Even-Odd Questions?  
 Everything about Factorials on the GMAT 
 Solving Complex Problems Using Number Line 
     Questions   
 Tough New Set: Number Properties Set 
 Power of a Number in a Factorial Problems 
 Number Properties Ability Quiz 
 Special Numbers and Sequences Problems 
 The E-GMAT Number Properties Knockout: 10 Great 700+ Ques for you! 
 Trailing Zeros Problems 
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

Hope it helps.

goldeneagle94 · Answer

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.

GMAT TIGER · Answer

Note: A square never has even number of distinct factors. Also the sum of distinct factors of a square is never even.

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

Suppose N = 4. It has 3 distinct factors: 1, 2 and 4. 
Suppose N = 9. It has 3 distinct factors: 1, 3 and 9. 
Suppose N = 16. It has 5 distinct factors: 1, 2, 4, 8, and 16.
Suppose N = 64. It has 7 distinct factors: 1, 2, 4, 8, 16, 32, and 64. 
But that not the case. In fact, the case is opposite. So it is sufficient because N is not a square.

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

If you follow the above pattern, you see 1 is always there. The sum of all distinct factors except 1 of N is even. If you add 1 on the even sum, that odd. So N is not a square. 
But that not the case. In fact, the case is opposite. So it is sufficient because N is again not a square.

So D.

To validate the above premises, lets assume if N = 10. Its factors are 1, 2, 5 and 10 and their sum = 18. So 10 has 4 distinct factors and 18, which is even, sum.

If N = 15. Its factors are 1, 3, 5 and 15 and their sum = 24. So 15 has 4 distinct factors and even sum of its distinct factors.

If N = 30. Its factors are 1, 2, 3, 5, 6, 10, 15 and 30 and their sum = 40. So 15 has 8 distinct factors and even sum of its distinct factors. HTH.

Note: It would be better if you do not post OA or spoiler before any response from other members.

cipher · Answer

1) The number of distinct factors of N is even.

Suppose N = 4. It has 3 distinct factors: 1, 2 and 4. 
Suppose N = 9. It has 3 distinct factors: 1, 3 and 9. 
Suppose N = 16. It has 5 distinct factors: 1, 2, 4, 8, and 16.
Suppose N = 64. It has 7 distinct factors: 1, 2, 4, 8, 16, 32, and 64. 
But that not the case. In fact, the case is opposite. So it is sufficient because N is not a square.

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

If you follow the above pattern, you see 1 is always there. The sum of all distinct factors except 1 of N is even. If you add 1 on the even sum, that odd. So N is not a square. 
But that not the case. In fact, the case is opposite. So it is sufficient because N is again not a square.

So D.

PS: A perfect square always have odd number of factors, for e.g a integer  n  and its square  n^2

Now,  n  will have always have even number of factors, (take any number and you will realise that factors come in pairs), now  n^2  will have all factors which  n  has + one more which is  n^2

Barkatis · Answer

If you don't know the tricks about perfect squares, it will take you a little time to demonstrate :

(1) SUFFICIENT: The factors of any number N can be sorted into pairs that multiply to give N. (For instance, the factors of 24 can be paired as follows: 1 and 24; 2 and 12; 3 and 8; 4 and 6.) However, if N is a perfect square, one of these ‘pairs’ will consist of just one number: the square root of N. (For example, if N were 49, it would have the factor pair 7 × 7) Since all of the other factors can be paired off, it follows that if N is a perfect square, then N has an odd number of factors. (If N is not a perfect square, then all of its factors can be paired off, so it will have an even number of factors.) This statement then implies that N is not a perfect square.

(2) SUFFICIENT: Let N be a perfect square. If N is odd, then all factors of N are odd. Therefore, by the above reasoning, N has an odd number of odd factors, so their sum must be odd. If N is even, let M be the product of all the odd prime factors (as many times as they appear in N – not distinct) of N, which is also a perfect square. (For instance, if N = 100, then M =5 × 5 = 25.) Then the sum of factors of M is odd, by the above reasoning. Furthermore, all other factors of N (i.e., that don’t also divide M) are even. The sum total is thus odd + even = odd. 
Therefore, the sum of the factors of any perfect square is odd, so this statement implies that N is not a perfect square. 
This statement can also be investigated by trying several cases of even perfect squares (4, 16, 36, 64, 100), noting that in each case the sums of the factors are odd, and generalizing.

The correct answer is D.

KarishmaB · Answer

Take an example N =  2^4 * 3^2 *5^4 
If we write down its factors, we get: 1, 2, 3, 4, 5, 6, 8, 10, 12..... total 75 factors.
Lets just consider the odd factors i.e. factors made by  3^2 *5^4  (including 1)
These will be a total of 3*5 = 15 odd factors.
When you add an odd number of odd factors, result will be an odd number.
The rest of the factors of N will be even i.e. they will include at least one 2. their sum will definitely be even because no matter how many even numbers you add, you always get an even result.
Adding an odd number to an even number, you will get an odd number. Hence, sum of all factors of a perfect square will always be odd.

Fijisurf · Answer

I do understand now.
However still not sure why perfect squares have "EVEN number of Even-factors". 
I understand that power of the only even prime factor "2" determines the number of even factors. So, if a number is a perfect square then all the powers will be even. Then by adding 1 to all powers we get odd numbers, which we multiply. Then the number of even factors will be odd.

KarishmaB · Answer

OK. If you understand the theory above, will you agree that total number of factors of a perfect square will be odd (e.g. 75 above)? Also, all factors will be either even or odd. Will you also agree that total number of odd factors of perfect square will be odd? See the e.g. above  3^2.5^4  will give 3*5 = 15 i.e. odd number of factors because powers are always even.
Then, the number of even factors should be Odd (75) - Odd(15) = Even(60).

KarishmaB · Answer

If you want the proper method: 
We will obtain an even factor by multiplying all the 15 odd factors we obtained above by  2 or 2^2 or 2^3 or 2^4 .
Therefore, we will obtain 15*4 = 60 even factors. (We can take 2 in four ways)
If the number is a perfect square, power of 2 will always be even. (We do not use 4 + 1 here because to make it an even number, there has to be at least one 2.)
Therefore, even factors of a perfect square will always be even.

DmitryFarber · Answer

You can see why this works if you build factor tables for a regular number and a perfect square:

30=
1*30=
2*15=
3*10=
5*6

Here all the factors come in pairs, so we will have an even number of factors.

36=
1*36=
2*18=
3*12=
4*9=
6*6
Here there is a pair of identical factors (that's the perfect square part), so we will always have an odd number.

Therefore, statement 1 tells us we don't have a perfect square. Our answer is "no"--sufficient.

As for the sum of the distinct factors, it's a bit more theoretical, but let's look at two possibilities:

1) The number is odd, in which case all of its factors are odd. If they come in pairs, each pair will add to make an even. (O+O=E) If the number is a perfect square, it will have an odd number of factors, leaving an extra odd number, which will make an odd sum. (E+O=O) Therefore, the sum of an odd perfect square's factors must be odd.
2) The number is even. If it is also a perfect square, it will have an even number of each prime factor, because the primes have to pair off (e.g. 100 = 10*10 = (2*5)(2*5)). All of its factors aside from 1 will be built from these primes, we will have an even number of evens (adding up to an even) and an even number of odds (also adding up to an even). Then 1 comes in and messes up all that harmony and makes the whole thing odd! Therefore, the sum of an even perfect square's factors must be odd.

So in either case, a perfect square's factors make an odd sum. Statement 2 gives an answer of "no," which is sufficient. The answer is D.

I hope this helps! You can definitely try this with some different numbers--it makes more sense when you play with a few examples.

glores1970 · Answer

I think we should mention that the unwritten assumption is that we are only talking about positive factors. If we include -ve factors, the number of factors of every integer is always even.

KarishmaB · Answer

Factors are positive numbers only. On the exam if you have 'a has 4 factors', it means it has 4 positive factors.

EvaJager · Answer

Thanks Bunuel for reminding us the very useful properties mentioned above.

For those interested, we can easily justify why the sum of the distinct divisors of a perfect square is odd.

If the number N is an odd perfect square, then all its divisors are odd. They come in pairs,  (1,N), (d_1, N/d_1), (d_2,N/d_2)... , except  \sqrt{N}  (we count it only once, its pair being itself). So, we have an odd number of odd divisors, whose sum will certainly be odd.

If N is an even perfect square, then N must be of the form  N=2^{2n}M , where M is an odd perfect square. All the odd divisors of N are the divisors of M, and as we have seen above, their sum (and number) is odd. All the even divisors of N are obviously even, so again, the sum of all the divisors is odd.

ShalabhAr · Answer

But in this particular question, it has not been specified if N is not equal to one.
Hence, N could be 1, perfect square, and have even number of factors namely 1 and -1.

Posted from my mobile device

ashish8 · Answer

How is B sufficient? Sum of distinct factors of a perfect square is odd, but if n is 2, then the sum is also odd.

Bunuel · Answer

The sum of distinct factors of a perfect square is ALWAYS ODD. (2) says that "the sum of all distinct factors of N is even", hence N is not a perfect square.

EvaJager · Answer

(2) states: The sum of all distinct factors of N is   even  .
Since the sum of distinct factors of a perfect square must be odd, we can conclude that N is not a perfect square.
So, the answer to the question "Is N a perfect square?" is a definite NO.
Therefore, (2) sufficient.

Not only perfect squares have the sum of their distinct factors odd. As you mentioned, for 2, the sum of its factors is odd, and it is not a perfect square.
So, if a number is a perfect square, then the sum of its factors is necessarily odd, but the reciprocal is not true. Meaning, if the sum of the factors is odd, the number is not necessarily a perfect square, it might be or not. But if the sum of the distinct factors is even, then certainly the number cannot be a perfect square.

Bunuel · Answer

Also check this:

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.

Hope it helps.

Is the positive integer N a perfect square? (1) The number

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

Is the positive integer N a perfect square? (1) The number

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards