S represents the set of positive integers that are factors of 7200. Ho

Question

S represents the set of positive integers that are factors of 7200. How many integers in S are multiples of 72?

A. 2
B. 5
C. 9
D. 10
E. 100

IanStewart · Accepted Answer

If 72x is a divisor of 7200, then 7200/72x is an integer, so, cancelling, 100/x is an integer. So x needs to be a divisor of 100. So that's the question - how many positive divisors does 100 have? If you know the method for counting divisors, you can get the answer quickly - prime factorize 100:

100 = 2^2 * 5^2

then look only at the exponents (2 and 2) : add 1 to each exponent, and multiply what you get. So 100 has (2 + 1)(2 + 1) = 3*3 = 9 divisors.

Or you could list all nine of the divisors quite easily: 1, 2, 4, 5, 10, 20, 25, 50, 100

Novice90 · Answer

7200 = (2^5)*(3^2)*(5^2)
72 = (2^3)*(3^2)

Multiple of 72 in set S can have 3 different powers of 2 (0,1,2)
Multiple of 72 in set S can have 3 different powers of 5 (0,1,2)

72 can be multiplied with 3 different powers of 2 and still be a factor of 7200: (2^0), (2^1), (2^2)
72 can be multiplied with 3 different powers of 5 and still be a factor of 7200: (5^0), (5^1), (5^2)

Therefore total combinations of multiples of 72, that are factors of 7200 is 3x3 = 9

TheKingInTheNorth · Answer

hi bunuel,

could you please post the solution .

Bunuel · Answer

I'd solve the same way as done by Ian Stewart here: https://gmatclub.com/forum/s-represents ... l#p1884546 Could you please point out what is unclear in that solution? Thank you.

goforgmat · Answer

shouldn't it be 10? why do we leave out 72 as a multiple ..

ganand · Answer

Hi  goforgmat ,

We are not leaving 72 as a multiple. Possible multiples are

72*1 = 72
72*2 = 144
72*4 = 288
72*5 = 360
72*10 = 720
72*20 = 1440
72*25 = 1800
72*50 = 3600
72*100 = 7200

The total is 9.

Thanks.

JeffTargetTestPrep · Answer

7200 = 72 x 100. Thus, 72m is a factor of 7200 and also a multiple of 72, as long as m is a factor of 100. Since 100 = 2^2 x 5^2, 100 has (2 + 1) x (2 + 1) = 9 factors. Thus, there are 9 multiples of 72 that are also factors of 7200.

Answer: C

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S represents the set of positive integers that are factors of 7200. Ho

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S represents the set of positive integers that are factors of 7200. Ho

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