Is 12,500,000 a factor of positive integer n?

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gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pmFollowed by: 8 members; Thanked: 11 times

by gmattesttaker2 » Sun Mar 30, 2014 9:22 pm

Hello,

Can you please tell me if my approach is correct here:

Is 12,500,000 a factor of positive integer N ?

(1) 5^8 is a factor of N
(2) 20^5 is a factor of N

OA: C

12,500,000 = 5^8 x 2^5. So is N/(5^8 x 2^5) an integer?

1) N/5^8 is an integer. I was wondering if this is sufficient since it means that N does not have 2^5.

2) 2^10 x 5^5 is a factor of N. I was wondering if this is sufficient as well since it means that N does not have 3 more 5's.

Can you please assist with this?

Thanks,
Sri

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Source: Beat The GMAT — Data Sufficiency | Sun Mar 30, 2014 9:22 pm

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[email protected]: Elite Legendary Member; Posts: 10392; Joined: Sun Jun 23, 2013 6:38 pmFollowed by: 511 members; Location: Palo Alto, CA; Thanked: 2867 times; GMAT Score:800

by [email protected] » Sun Mar 30, 2014 10:28 pm

Hi Sri,

You've prime-factored 12,500,000 correctly. Now you have to be clear on what the question asks for.

For 12,500,000 to be a factor of N, N MUST contain 5^8 AND 2^5 (it can include other primes as well, but must have the aforementioned primes to be divisible by 12,500,000). This is a YES/NO question.

Fact 1: 5^8 is a factor of N.

This doesn't give us the necessary information to answer the question.
IF N = 5^8, then the answer to the question is NO
IF N = 5^8(2^5), then the answer to the question is YES.
Fact 1 is INSUFFICIENT

Fact 2: 20^5 is a factor of N.

This breaks down into (2^10)(5^5)

IF N = (2^10)(5^5) then the answer to the question is NO (in basic terms, there are not "enough 5s")
IF N = (2^10)(5^5)(5^3) then the answer to the question is YES
Fact 2 is INSUFFICIENT

Combined, we know that the following numbers are factors of N:
5^8
(2^10)(5^5)

This means that N MUST consist of at least (5^8) and (2^10). This information is enough to confirm that the answer is YES.
Combined, SUFFICIENT

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon Mar 31, 2014 9:49 am

gmattesttaker2 wrote:Is 12,500,000 a factor of positive integer N ?

(1) 5^8 is a factor of N
(2) 20^5 is a factor of N

I'd like to elaborate a little on Rich's excellent post.

For questions involving factors (aka "divisors"), we can say:
If k is a divisor of N, then k is "hiding" within the prime factorization of N
Consider these examples:
3 is a divisor of 24 because 24 = (2)(2)(2)(3), and we can clearly see the 3 hiding in the prime factorization.
Likewise, 5 is a divisor of 70 because 70 = (2)(5)(7)
And 8 is a divisor of 112 because 112 = (2)(2)(2)(2)(7)
And 15 is a divisor of 630 because 630 = (2)(3)(3)(5)(7)

So...

Target question: Is 12,500,000 a factor of positive integer N ?

12,500,000 = (2)(2)(2)(2)(2)(5)(5)(5)(5)(5)(5)(5)(5)
Or we can write: 12,500,000 = (2^5)(5^8)

Let's rephrase the target question as...
REPHRASED target question: Are there five 2's and eight 5's hiding in the prime factorization of N?

Statement 1: 5^8 is a factor of N
This tells us that there are eight 5's hiding in the prime factorization of N.
What about the five 2's?
We can't say whether or not the five 2's are also hiding in the prime factorization of N.
As such, statement 1 is NOT SUFFICIENT

Statement 2: 20^5 is a factor of N
Notice that 20 = (2)(2)(5) = (2^2)(5)
So, 20^5 = (2^10)(5^5)
In other words, (2^10)(5^5) is a factor of N.
In other words, there are ten 2's and five 5's hiding in the prime factorization of N.
So, we can't say whether or not there are five 2's and eight 5's hiding in the prime factorization of N.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 guarantees that we have eight 5's hiding in the prime factorization of N.
Statement 2 guarantees that we have five 2's hiding in the prime factorization of N.
So, the statements combined guarantee that there are five 2's and eight 5's hiding in the prime factorization of N
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pmFollowed by: 8 members; Thanked: 11 times

by gmattesttaker2 » Wed Apr 02, 2014 5:26 pm

[email protected] wrote:Hi Sri,

You've prime-factored 12,500,000 correctly. Now you have to be clear on what the question asks for.

For 12,500,000 to be a factor of N, N MUST contain 5^8 AND 2^5 (it can include other primes as well, but must have the aforementioned primes to be divisible by 12,500,000). This is a YES/NO question.

Fact 1: 5^8 is a factor of N.

This doesn't give us the necessary information to answer the question.
IF N = 5^8, then the answer to the question is NO
IF N = 5^8(2^5), then the answer to the question is YES.
Fact 1 is INSUFFICIENT

Fact 2: 20^5 is a factor of N.

This breaks down into (2^10)(5^5)

IF N = (2^10)(5^5) then the answer to the question is NO (in basic terms, there are not "enough 5s")
IF N = (2^10)(5^5)(5^3) then the answer to the question is YES
Fact 2 is INSUFFICIENT

Combined, we know that the following numbers are factors of N:
5^8
(2^10)(5^5)

This means that N MUST consist of at least (5^8) and (2^10). This information is enough to confirm that the answer is YES.
Combined, SUFFICIENT

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

Hello Rich,

Thank you very much for your excellent and detailed explanation.

Best Regards,
Sri

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gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pmFollowed by: 8 members; Thanked: 11 times

by gmattesttaker2 » Wed Apr 02, 2014 5:27 pm

Brent@GMATPrepNow wrote:
gmattesttaker2 wrote:Is 12,500,000 a factor of positive integer N ?

(1) 5^8 is a factor of N
(2) 20^5 is a factor of N
I'd like to elaborate a little on Rich's excellent post.

For questions involving factors (aka "divisors"), we can say:
If k is a divisor of N, then k is "hiding" within the prime factorization of N
Consider these examples:
3 is a divisor of 24 because 24 = (2)(2)(2)(3), and we can clearly see the 3 hiding in the prime factorization.
Likewise, 5 is a divisor of 70 because 70 = (2)(5)(7)
And 8 is a divisor of 112 because 112 = (2)(2)(2)(2)(7)
And 15 is a divisor of 630 because 630 = (2)(3)(3)(5)(7)

So...

Target question: Is 12,500,000 a factor of positive integer N ?

12,500,000 = (2)(2)(2)(2)(2)(5)(5)(5)(5)(5)(5)(5)(5)
Or we can write: 12,500,000 = (2^5)(5^8)

Let's rephrase the target question as...
REPHRASED target question: Are there five 2's and eight 5's hiding in the prime factorization of N?

Statement 1: 5^8 is a factor of N
This tells us that there are eight 5's hiding in the prime factorization of N.
What about the five 2's?
We can't say whether or not the five 2's are also hiding in the prime factorization of N.
As such, statement 1 is NOT SUFFICIENT

Statement 2: 20^5 is a factor of N
Notice that 20 = (2)(2)(5) = (2^2)(5)
So, 20^5 = (2^10)(5^5)
In other words, (2^10)(5^5) is a factor of N.
In other words, there are ten 2's and five 5's hiding in the prime factorization of N.
So, we can't say whether or not there are five 2's and eight 5's hiding in the prime factorization of N.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 guarantees that we have eight 5's hiding in the prime factorization of N.
Statement 2 guarantees that we have five 2's hiding in the prime factorization of N.
So, the statements combined guarantee that there are five 2's and eight 5's hiding in the prime factorization of N
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent

Hello Brent,

Thank you very much for your detailed and excellent explanation with the examples.

Best Regards,
Sri

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