What is a perfect square?

[trall3]

Hi, I'm confused here. I've tried searching but different definitions keep showing up.

One def: Squares of integers

Another: Squares of primes:
Which alwas have exactly tree factors (that I understand)

Also, I've read that the ONLY integers with exactly 3 factors are squares of primes, but what about 1*2*5=10 and 1*2*3=6???

Thank yo

[GMATBootcamp]

Hi trall3,

By definition a perfect square is any variable x, where the square root of x yields a whole number.

eg. 25, 16, 9 are all perfect squares because the square roots of these numbers are whole numbers.

perfect squares can also apply to terms:

eg. the perfect square of x^2+2x+1 is x+1


To answer your other question, the majority of numbers have an even number of factors.

eg. the number 10 has distinct 4 factors

1, 2, 5, 10

But if a problem statement tells us that a number x has a square root and an odd number of factors, then the number x must be the square of a prime number.

eg.

if x = 25

the square root of 25 is 5

distinct factors of 25 are 1, 5, and 25.

(in your question 10 and 6 aren't squares of primes, so your last statement would not be true)

Hope that helps!

[cbenk121]

A perfect square is a square of an integer.

For example, 4 is a perfect square.

An easy way to determine if a number is a perfect square, is to take the square root of it. Does it come out clean (i.e. no decimal)? Thinking of this way, you can quickly determine that 81 is a perfect square, but 80 is not.

[cbenk121]

To answer your other question, the majority of numbers have an even number of factors.

eg. the number 10 has distinct 4 factors

1, 2, 5, 10

But if a problem statement tells us that a number x has a square root and an odd number of factors, then the number x must be the square of a prime number.

eg.

if x = 25

the square root of 25 is 5

distinct factors of 25 are 1, 5, and 25.

Did not know that...what else can you draw if a number has an odd number of factors, besides that it could be a perfect square?

[tgf]

Hi trall3,


But if a problem statement tells us that a number x has a square root and an odd number of factors, then the number x must be the square of a prime number.

Hope that helps!

Thank you, that was helful. But one last question:

Suppose we know a number has an odd number of factors: Don't we need to know if it's a a odd number of _distinct_ factors?

For example 1*7*7=49 Has four factors if the 7 is counted twice...? Isn't that the defition of factors? (total number of terms, even those occuring multiple number of times)?

Thank you

[tgf]

about the username confusement. I was at school when posting the first question, and had forgotten my password so I had to register again. When I replied this morning I did if from my laptop, and it automatically logged me on with my first username... Sorry for the confusion!

[GMATBootcamp]

Distinct factors actually means the total number unique factors. (if that helps you interpret things at all)


using your example, the number 49 has 3 distinct factors
1, 7, and 49

The number 7 shouldn't be counted twice because the number 49, which is 7^2 is also considered a factor.

If you're still confused, try doing a prime factorization tree

If we perform prime factorization on 49 we get 7 and 7

Use the prime factors to determine all possible factors of the number 49 (remember to include the number 1)

The factors of 49 are again: 1, 7, 49



HTH!

[ChrisHinkle]

Very important here to distinguish between the number of factors and the number of prime factors. When you do the prime factorization, you often will keep duplicates, so the prime factorization of 49 is 7*7. But if we are just listing all factors, list each one once.

It is true that if a number has an odd number of distinct positive factors, the number must be a perfect square. Otherwise, each factor pairs up with another to equal the original number.

[beastly B]

its pretty simple (for example) 16 is a perfect square because 4*4=16
hope it works
I BEAT THE GMAT LOL

[beastly B]

i need a good (not complicated) strategy think someone can help?

[Tani]

Every integer greater than one (including primes) has at least two factors: one and itself.

To have three factors total, the number must have one, itself and only one more = total three.

The only way this is possible is if the number is a perfect square of a prime.

Perfect squares of non-primes (e.g. 16) will still have an odd number of factors because of the repeated square root.