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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

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

yangsta8 wrote:

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

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.

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

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.

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, 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.

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.
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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?

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?

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.
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