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# Finding Number of Factors of an integer - question?

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Finding Number of Factors of an integer - question? [#permalink]  27 Nov 2013, 05:52
Hi,

Can anyone please help me with the formula for finding the total no. of factors of an integer?

The usual way is prime factorization; then increasing the exponent of each prime factor by one; and multiplying the raised numbers.

E.g:

Q. Find the no. of factors of 90?

A.

90 = 2^1 x 3^2 x 5^1

So total factors = (1+1) x (2+1) x (1+1) = 2 x 3 x 2 = 12.

I have a test prep where solution actually doubles this answer saying that 12 are positive factors and there are as many negative factors.

So the total no. of factors should be 12 x 2 = 24.

But I researched a bit on the internet and almost everywhere they don't double the answer, i.e. they say answer is 12.

What is right on GMAT?

Thanks,
federite
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Re: Finding Number of Factors of an integer - question? [#permalink]  27 Nov 2013, 06:05
You are correct - there are 12 factors in the number 90.

Generally, you do not include negative numbers. I'd be curious as to the actual wording of the question of the test prep.
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Re: Finding Number of Factors of an integer - question? [#permalink]  27 Nov 2013, 06:13
Thanks for the reply.
The actual wording was what is the total number of factors of (a positive integer). I don't remember the exact number though.

Adding the negative number of factors sounds logical though.

Not sure whether I should do that. Fyi - The GMAT Club Math book doesn't add the negative factors. So doesn't MGMAT strategy guide.

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Re: Finding Number of Factors of an integer - question? [#permalink]  27 Nov 2013, 06:19
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If it's just "find total number of factors of x", then do not include negative numbers. Factors should be positive, as prime numbers are. Factors are essentially prime factorization.
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Re: Finding Number of Factors of an integer - question? [#permalink]  27 Nov 2013, 06:26
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Expert's post
federite wrote:
Hi,

Can anyone please help me with the formula for finding the total no. of factors of an integer?

The usual way is prime factorization; then increasing the exponent of each prime factor by one; and multiplying the raised numbers.

E.g:

Q. Find the no. of factors of 90?

A.

90 = 2^1 x 3^2 x 5^1

So total factors = (1+1) x (2+1) x (1+1) = 2 x 3 x 2 = 12.

I have a test prep where solution actually doubles this answer saying that 12 are positive factors and there are as many negative factors.

So the total no. of factors should be 12 x 2 = 24.

But I researched a bit on the internet and almost everywhere they don't double the answer, i.e. they say answer is 12.

What is right on GMAT?

Thanks,
federite

Factor is a "positive divisor" (at least on the GMAT). So, for example the factors of 4 are 1, 2, and 4 ONLY.

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.

For more check: math-number-theory-88376.html

Questions to practice:
how-many-odd-positive-divisors-does-540-have-106082.html
how-many-factors-does-36-2-have-126422.html
how-many-different-positive-integers-are-factor-of-130628.html
how-many-distinct-positive-factors-does-30-030-have-144326.html
m02-72467.html

Hope this helps.
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Re: Finding Number of Factors of an integer - question? [#permalink]  28 Nov 2013, 04:37
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Definition from Manhattan GMAT, page 15 of Number Properties strategy guide:
"A factor is a positive integer that divides evenly into an integer". So, at least in the GMAT world, consider only positive factor.
Factor and divisor means the same thing, and from Wikipedia:
"Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only the positive ones (1, 2, and 4) would usually be mentioned."
So in the Math world you should consider also negative ones. But this is not the case.
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Re: Finding Number of Factors of an integer - question? [#permalink]  08 Jan 2015, 20:49
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Re: Finding Number of Factors of an integer - question?   [#permalink] 08 Jan 2015, 20:49
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# Finding Number of Factors of an integer - question?

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