Which of the following is not a perfect square

Question

1. 100856
2. 325137
3. 945729
4. All of these

Hint- value of perfect sq has to end in 1,4,5,6,9...so option 2 is definately not a perfect square

But how to find the rest...is ther any shortcut or will we have to find the prime factors of all n bring them to standard form

yangsta8 · Answer

A perfect square will have exactly an even number of each of its factors. 
81 for example:
81 = 9x9 = 3x3x3x3 which is essentially 2 sets of 3x3
Or:
36 = 6x6 = 2x3x2x3 which is 2 sets of 2x3

1. 100856
2. 325137
3. 945729
4. All of these

You mention option 2 is not correct, so neither is option 4.
Statement 1: 100856
You can only factor it by 2 three times. This is not an even amount of 2s so it cannot be a perfect square.
Statement 3: 945729
You can factor 3 three times. This is not an even amount of 3s so it cannot be a perfect square

So if I’m not wrong then none of these are perfect square? Which seems odd as that’s not an answer option. Or I could be wrong 

virtualanimosity · Answer

Ans is 
optn 4 as it says 'all of these' meaning all of these r not perfect squares

yangsta8 · Answer

Ahh yes you are right I didn't read it properly. Also to answer your original question I guess there is no need to factor all prime factors, just factor a few until you find that there are not an even number.

shalva · Answer

A simple point, but worth remembering, Kudos!

Bunuel · Answer

I think the red part needs some clarification:

A perfect square will have exactly an even number of each of its factors.

That's not true.

I think you meant that perfect square n^2, has even number of prime factors of n or in other words prime factors of n have even powers.

In your example: 81=3*3*3*3=3^4, but 81 has the factor 27, what about it? There are three 27 in 81.
Or 36=6*6=2^2*3^2 but what about 12 which is factor of 36? There are three 12 in 36.

There are some  tips about the perfect square  though:

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

Hope it helps.

shalva · Answer

yeah, it holds only for prime factors

shalva · Answer

Man, there should be a way to add some posts to favorites, to "memories" or something like that. The only method available is to save permalinks . . .

yangsta8 · Answer

Yes you are right. Thanks for the clarification

nonameee · Answer

Bunuel, is the concept of a perfect square tested on the GMAT? I have come across it several times on this forum, but I am not sure if it is tested on the GMAT. Thank you.

Bunuel · Answer

What do you mean by "the concept of a perfect square"??? A perfect square, is just an integer that can be written as the square of some other integer, for example 16=4^2, is a perfect square.

nonameee · Answer

What I mean is that I've come across several questions here on the forum that have a phrase "perfect square", and I have never heard of it before. So the reason I'm asking is if I have to pay attention to these questions (and read about some useful properties of perfect squares) or not?

Bunuel · Answer

Again: a perfect square, is just an integer that can be written as the square of some other integer, for example 16=4^2, is a perfect square. So perfect square  is just a name  of such integers as 1, 4, 9, 16, ... There are several properties of a perfect square, which might be useful while solving specific GMAT questions. Now, it's up to you to decide whether to study these question and properties or not.

nonameee · Answer

OK. Thanks. I guess I have to study it.

maxim7777 · Answer

Could you please prove these rules?

I know only that the total number of factors (of a perfect square) is odd. Could you clarify the others?

Thanks in advance.

JeffTargetTestPrep · Answer

We know that a perfect square must end with 0, 1, 4, 5, 6, or 9, so we can see choice 2 is not a perfect square.

Now let’s check choice 1. Since 310^2 = 96,100 and 320^2 = 102,400, then, if 100,856 is a perfect square, it must be the square of either 314 or 316 since either one of these integers will produce a units digit of 6 when squared. However, since 314^2 = 98,596 and 316^2 = 99,856, we see that choice 1 is a not a perfect square, either.

Since both choices 1 and 2 are not perfect squares, we will conclude that the correct answer must be choice 4 (even though we have not verified that choice 3 is not a perfect square).

Answer: 4

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Which of the following is not a perfect square

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Which of the following is not a perfect square

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