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GMAT Number properties - perfect squares

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GMAT Number properties - perfect squares [#permalink] New post 05 May 2011, 18:27
Hello Brains,
I recently made a few points about perfect squares that might be helpful to solve problems quickly.

These are the props :
A perfect square has an even number of powers of prime factors
Any perfect square integer always has an odd number of distinct factors.
For any perfect square, the sum of its distinct factors is always odd
Now for the substantiations:
A perfect square has an even number of powers of prime factors
This is "THE BASIC RULE" and the other two (and perhaps many others) rules can be derived out of this rule. The rule says - For a perfect square, N, if N is prime factorized, say N = (px) * (qy), x and y can and will only be even integers. This seems pretty obvious. If x and/or y were infact odd, there wouldn't be able to find sqrt(N) in an even positive integer.

With that now settled, in order to prove the second point,

Any perfect square integer always has an odd number of distinct factors.
there is this tiny hack that lets you find the number of distinct factors a number has. it is a simple two step process.
Factor the number into its prime components, N = (px) * (qy)
The number of distinct factors =(x+1)*(y+1).
Since we proved just now that for such a perfect square x and y will be even,
Code:
the number of factors = (even + 1)*(even + 1)
                                    = odd* odd
                                    = odd

The third rule is a bit tricky

For any perfect square, the sum of its distinct factors is always odd
I havent arrived at a thorough proof for this one as of now. But one thing to remember is that any perfect square will have an odd number of odd factors and an even number of even factors. So adding all these up, we get an odd integer. Try it.
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Re: GMAT Number properties - perfect squares [#permalink] New post 07 May 2011, 23:37
May be this snippet from the GMAT Club Math Book wil help you deduce the proof.

math-number-theory-88376.html


Finding the Number of Factors of an Integer

First make prime factorization of an integer , where , , and are prime factors of and , , and are their powers.

The number of factors of will be expressed by the formula . NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450:

Total number of factors of 450 including 1 and 450 itself is factors.


Finding the Sum of the Factors of an Integer

First make prime factorization of an integer , where , , and are prime factors of and , , and are their powers.

The sum of factors of will be expressed by the formula:

Example: Finding the sum of all factors of 450:

The sum of all factors of 450 is
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Re: GMAT Number properties - perfect squares   [#permalink] 07 May 2011, 23:37
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