Let f(n) = the number of distinct factors n has. For example, f(20) =

Question

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.

chetan2u · Accepted Answer

Hi, You don't have to do any calculations but require to know the property of factors.. A SQUARE has odd number of factors... So 225 is ODD.. Now check the choices.. Only D is multiple of two squares so the number of factors will be odd*odd=odd.. Rest all choices will be EVEN 100=2^2*5^2.. Number of factors= (2+1)(2+1)=3*3.. 10000=2^4*5^4.. Number of factors=(4+1)(4+1)=25.. answer would be F(100).F(10000)=9*25=225.. Ans D :-D

BrentGMATPrepNow · Answer

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.

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 
 https://www.youtube.com/watch?v=lVZ9GXd6ZBE

https://www.youtube.com/watch?v=W72A4b05K9Y

Nanobotstv · Answer

Thanks for reply. I am still struggling to understand the logic behind this. Can you elaborate more on this?

chetan2u · Answer

Hi..

Each number can be written as sum of two factors..
Example 10..1*10;2*5;.
But a square has one set which has same integer, so only one number comes out of that set..that is why ODD.
Example 36...1*36;2*18;3*12:4*9;6*6.... here 6*6 gives only one integer 6

Joc456 · Answer

The easiest way to do this is to use the trick of breaking down everything into primes then looking at the exponents. Lets take 100 as an example

100 breaks down to 10*10 which is 2*5*2*5 this is 2²*5². The exponent trick is to break down a number into primes and then add one to each exponent, the multiply. So this would result in 3*3 or 9 factors.

If you do this for the numbers in the question you'll see 100 has 9 factors, 1000 has 16 factors, and and 100000 has 25 factors (these are all perfect squares).

So 9*25=225 hence D

ganand · Answer

Any integer can be written as a product of prime numbers.

If  N = p^{a} q^{b} r^{c} , then no. of factors of  N = (a+1) 	imes (b+1) 	imes (c+1) .

10 = 2^{1} 	imes 5^{1} , No. of factors = (1+1)*(1+1) = 4
 100 = 2^{2} 	imes 5^{2} , No. of factors = (2+1) * (2+1) = 9
 1000 = 2^{3} 	imes 5^{3} , No. of factors = (3+1) * (3+1) = 16
 10000 = 2^{4} 	imes 5^{4} , No. of factors = (4+1) * (4+1) = 25

Option (d) is the correct answer.

EMPOWERgmatRichC · Answer

Hi All,

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

rahul16singh28 · Answer

10 - 4 Factors (So, Option A & E ruled out).
100 - 9 Factors
1000 - 16 Factors (Option C & B ruled out).

Hence, D.

mimajit · Answer

This is how i solved

10= 2^1 * 5^1 , No. of divisors = (1+1)*(1+1) = 4
100=2^2 * 5^2, No. of divisors = (2+1) * (2+1) = 9
1000=2^3*5^3, No. of divisors = (3+1) * (3+1) = 16
10000=2^4 * 5^4 No. of divisors = (4+1) * (4+1) = 25

target question Which of the following products is equal to 225?

=25*9 = 225 ie F( 10000)* f ( 100 ) ans choice d

bumpbot · Answer

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Let f(n) = the number of distinct factors n has. For example, f(20) =

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