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There are 3 ways to make the number 12 using products of two

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by Max@Math Revolution » Tue Jul 24, 2018 12:17 am

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[Math Revolution GMAT math practice question]

There are 3 ways to make the number 12 using products of two positive integers. These are 1*12, 2*6, and 3*4. In how many ways can 2700 be written as the product of two positive integers?

A. 6
B. 12
C. 15
D. 18
E. 36
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by GMATGuruNY » Tue Jul 24, 2018 2:59 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

There are 3 ways to make the number 12 using products of two positive integers. These are 1*12, 2*6, and 3*4. In how many ways can 2700 be written as the product of two positive integers?

A. 6
B. 12
C. 15
D. 18
E. 36
To count the factors of a positive integer:
1. Prime-factorize the integer
2. Write the prime-factorization in the form (a^p)(b^q)(c^r)...
3. The number of factors = (p+1)(q+1)(r+1)..

2700 = 2²3³5²

Adding 1 to each exponent and multiplying, we get:
Total number of factors = (2+1)(3+1)(2+1) = 36

These 36 factors can be used to form 18 FACTOR PAIRS, as follows:
1*2700
2*1350
3*300
And so on.

The correct answer is D.
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by Max@Math Revolution » Thu Jul 26, 2018 12:33 am
=>

2700 = 2^2*3^3*5^2
The number of distinct factors of 2700 is (2+1)(3+1)(2+1) = 36.
Since the order of multiplication does not matter (i.e. 30 * 90 = 90*30), the number of pairs of positive integers that multiply to give 2700 is 36/2 = 18.

Therefore, the answer is D.
Answer: D
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