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A Factor Question, Can Anybody Explain with Details*

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Re: A Factor Question, Can Anybody Explain with Details*

by parallel_chase » Thu Sep 04, 2008 2:58 pm
thegmatbeater wrote:2
How many factors does 36 have?
2
8
24
25
26
Are you sure about the answer choices.

36 has 9 factors

36 = 2*2*3*3 = 2^2 * 3^2
Now only look at the powers of the prime factors.
(2+1)*(2+1) = 3*3 = 9

If you are asking about prime factors, then there are 2 prime factors, 2 and 3.
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by mehravikas » Thu Sep 04, 2008 8:05 pm
36 will have 9 factors -

1 * 36
6 * 6
12 * 3
9 * 4
18 * 2

Unique factors - 1, 2, 3, 4, 6, 9, 18, 36
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by P_mashru » Thu Sep 04, 2008 9:13 pm
Dear thegmatbeater!

You might have taken this question from MGMAT. The correct question ask no of fectors of 36^2 and not 36.

Pl confirm and I will explain why correct answer is 25
I am I
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by shahdevang87 » Fri Sep 05, 2008 7:53 pm
Hi,

Can you please explain how do we het 25? I read the explanation from MGMAT. But could not understand that. They are mentioning following.

Would you please explain why we are doing 5*5 ? Means would we not be having some repeations if we do that?

------------------------------------------------------------------
36^2 can be expressed as the product of its prime factors, raised to the appropriate exponents:

362 = (22 × 32)2 = 24 × 34

So, the prime box of 362 contains four 2's and four 3's, as shown:

2 2 2 2 3 3 3 3



Now, if you pick any combination of these primes and multiply them all together, the product will be a factor of 362. As you take primes from this prime box to construct a factor of 362, note that you can choose up to four 2's and up to four 3's. In fact, you have FIVE choices for the number of 2's you put into the factor: zero, one, two, three, or four 2's. Likewise, you have the same FIVE choices for the number of 3's you put into the factor: zero, one, two, three, or four. (It doesn't matter what order you pick the factors, since order doesn't matter in multiplication.)

Note that you are allowed to pick zero 2's and zero 3's at the same time. By doing so, you are constructing the factor 20 × 30 = 1, which is a separate, valid factor of 362.

Since you have five independent choices for the number of 2's you pick AND you have five independent choices for the number of 3's you pick, you MULTIPLY the number of choices together to get the number of options you have overall. Thus you have 5 × 5 = 25 different ways to construct a factor. This means that there are 25 different factors of 362.

The correct answer is D.
-------------------------------------------------------------------------------

Thanks in advance.


[

quote="P_mashru"]Dear thegmatbeater!

You might have taken this question from MGMAT. The correct question ask no of fectors of 36^2 and not 36.

Pl confirm and I will explain why correct answer is 25[/quote]
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by cramya » Fri Sep 05, 2008 8:36 pm
I factored 1296 in to

2*2*2*2*3*3*3*3 = 8 factors so far

1 is always a factor so 9

Out of 2*2*2*2*3*3*3*3 there are 4 ways(lets say m) to choose 2(2,22,222,2222) and 4 ways(lets say n ways) to choose 3 (3,33,333,3333) so they can both be combined m*n ways = 16

Total factors = 9 + 16 = 25
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by cramya » Fri Sep 05, 2008 8:54 pm
Addition to my earlier post:

2*2*2*2*3*3*3*3 = 8 factors

When I said 8 factors I meant these below:

2 3
22 33
222 333
2222 3333
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by thegmatbeater » Sat Sep 06, 2008 3:05 pm
Yes, the number I tried to write was 36^2..Can you explain the answer?
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by parallel_chase » Sun Sep 07, 2008 2:41 am
thegmatbeater wrote:Yes, the number I tried to write was 36^2..Can you explain the answer?
36^2

= 6*6*6*6

= 2^4 * 3^4

Only look at the exponents

(4+1) * (4+1)

you add 1 to the power of each prime factor and multiply

5*5 = 25

Hope its clear.
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by Ian Stewart » Sun Sep 07, 2008 8:14 am
Yes, to count the divisors of a positive integer, you can:

-prime factorize completely;
-ignore the primes, and look at the exponents;
-add one to each exponent and multiply.

You can actually do the above question without doing any work. Notice that every perfect square has only even exponents in its prime factorization. When you add one to each exponent and multiply, you will be multiplying only odd numbers- that is, every perfect square has an odd number of divisors. Since 36^2 is clearly a perfect square, it must have an odd number of divisors, and 25 is the only odd answer choice.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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