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Let f(n) = the number of distinct factors n has. For example

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Let f(n) = the number of distinct factors n has. For example

by alanforde800Maximus » Tue Oct 11, 2016 5:01 am
Let f(n) = the number of distinct factors n has. For example, f(20) = 6, because 20 has six factors(1,2,4,5,10 and 20). Which of the following products is equal to 225?

a) f(10).f(100)
b) f(100).f(1000)
c) f(1000).f(10000)
d) f(100).f(10000)
e) f(10).f(1000)

Please assist with above problem.
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by Brent@GMATPrepNow » Tue Oct 11, 2016 7:28 am
I'm assuming that the periods in your post are meant to represent multiplication.
I have replaced the periods with "x"

Please go back and edit your original post.
alanforde800Maximus wrote:Let f(n) = the number of distinct factors n has. For example, f(20) = 6, because 20 has six factors(1,2,4,5,10 and 20). Which of the following products is equal to 225?

a) f(10) x f(100)
b) f(100) x f(1000)
c) f(1000) x f(10000)
d) f(100) x f(10000)
e) f(10) x f(1000)
NICE RULE
If N = (p^a)(q^b)(r^c)..., where p, q, r,...(etc.) are prime numbers, then the total number of positive divisors of N is equal to (a+1)(b+1)(c+1)...

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) = (5)(4)(2) = 40

----------------------------------------

Let's test a few values:
100 = (2^2)(5^2)
So, the number of positive divisors of 100 = (2+1)(2+1) = (3)(3) = 9
So, f(100) = 9

1000 = (2^3)(5^3)
So, the number of positive divisors of 1000 = (3+1)(3+1) = (4)(4) = 16
So, f(1000) = 16

10000 = (2^4)(5^4)
So, the number of positive divisors of 10000 = (4+1)(4+1) = (5)(5) = 25
So, f(10000) = 25

Which of the following products is equal to 225?
225 = 9 x 25
= f(100) x f(10000)

Answer: D

RELATED RESOURCES
- Counting the divisors of large number: https://www.gmatprepnow.com/module/gmat ... /video/828
- Prime factorization: https://www.gmatprepnow.com/module/gmat ... /video/825
Brent Hanneson - Creator of GMATPrepNow.com
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by [email protected] » Tue Oct 11, 2016 9:27 am
Hi alanforde800Maximus,

In this prompt, you can use "design" of this question, and the answer choices, to your advantage. We're told to find a product of two values that equals 225. In this prompt, there aren't that many ways to get to that product in this way though:

1 x 225
5 x 45
9 x 25
15 x 15

Looking at the answer choices, we know that each of those numbers has MORE than 1 factor, so (1x225) is out. We also know that none of those answers is the product of the same term twice, so (15x15) is out.

f(10) = 1,2,5,10 = 4 terms - which isn't an option, so we can eliminate Answers A and E. f(100) clearly has more than 5 factors, so we're looking for an answer that is (9x25).

From here, you just have to factor down two of the three numbers: 100, 1000, 10000 - you'll either have the exact math OR you'll have one of the correct terms and the remaining term will be the other one that you need. Thus, you'll know which answer is the match.

Final Answer: D

GMAT assassins aren't born, they're made,
Rich
Contact Rich at [email protected]
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