## Confused--Need help here

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### Confused--Need help here

by dddanny2006 » Mon Nov 11, 2013 12:40 pm
Is mn divisible by 40?

1.m is divisible by 10
2.n is divisible by 4

I saw this problem on Vimeo.Can anyone tell me again why 1 and 2 are insufficient.I want to understand the Prime Box method here.

Thanks

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by Brent@GMATPrepNow » Mon Nov 11, 2013 1:30 pm
dddanny2006 wrote:Is mn divisible by 40?

1) m is divisible by 10
2) n is divisible by 4
SOME BACKGROUND: A lot of integer property questions can be solved using prime factorization.
For questions involving divisibility, divisors, factors and multiples, we can say:
If N is divisible by k, then k is "hiding" within the prime factorization of N

Examples:
24 is divisible by 3 <--> 24 = (2)(2)(2)(3)
70 is divisible by 5 <--> 70 = (2)(5)(7)
330 is divisible by 6 <--> 330 = (2)(3)(5)(11)
56 is divisible by 8 <--> 56 = (2)(2)(2)(7)

Okay, now onto the question at hand . . .

Target question: Is mn divisible by 40?

Since 40 = (2)(2)(2)(5), we can rephrase the target question as . . .

REPHRASED target question: Are there three 2's and one 5 hiding in the prime factorization of mn?

Statement 1: m is divisible by 10
There are several values of m and n that satisfy this condition. Here are two possible cases:
Case a: m = 10 and n = 8, in which case mn is divisible by 40
Case b: m = 10 and n = 3, in which case mn is not divisible by 40
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: n is divisible by 4
There are several values of m and n that satisfy this condition. Here are two possible cases:
Case a: m = 10 and n = 4, in which case mn is divisible by 40
Case b: m = 3 and n = 4, in which case mn is not divisible by 40
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1: Since 10 = (2)(5), statement 1 tells us that there is one 2 and one 5 hiding in the prime factorization of m.
Statement 2: Since 4 = (2)(2), statement 2 tells us that there are two 2's hiding in the prime factorization of n.
So, there must be three 2's and one 5 hiding in the prime factorization of mn

Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Cheers,
Brent
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by Brent@GMATPrepNow » Mon Nov 11, 2013 1:34 pm
dddanny2006 wrote:Is mn divisible by 40?

1. m is divisible by 10
2. n is divisible by 4
Here's one more approach . ..

Target question: Is mn divisible by 40?

Statement 1: m is divisible by 10
There are several values of m and n that satisfy this condition. Here are two possible cases:
Case a: m = 10 and n = 8, in which case mn is divisible by 40
Case b: m = 10 and n = 3, in which case mn is not divisible by 40
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: n is divisible by 4
There are several values of m and n that satisfy this condition. Here are two possible cases:
Case a: m = 10 and n = 4, in which case mn is divisible by 40
Case b: m = 3 and n = 4, in which case mn is not divisible by 40
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1: If m is divisible by 10, then m = 10k (for some integer k)
Statement 2: If n is divisible by 4, then n = 4j (for some integer j)
So, mn = (10k)(4j) = 40kj
As we can see, mn is a multiple of 40, which means mn is definitely divisible by 40
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Cheers,
Brent
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by dddanny2006 » Mon Nov 11, 2013 1:51 pm
Hey Brent,

Can you explain this problem in terms of
Statement one tells us --2 x 5...

Statement two tells us --2 x 2...

I want to prove insufficieny based on the above concept.

Brent@GMATPrepNow wrote:
dddanny2006 wrote:Is mn divisible by 40?

1) m is divisible by 10
2) n is divisible by 4
SOME BACKGROUND: A lot of integer property questions can be solved using prime factorization.
For questions involving divisibility, divisors, factors and multiples, we can say:
If N is divisible by k, then k is "hiding" within the prime factorization of N

Examples:
24 is divisible by 3 <--> 24 = (2)(2)(2)(3)
70 is divisible by 5 <--> 70 = (2)(5)(7)
330 is divisible by 6 <--> 330 = (2)(3)(5)(11)
56 is divisible by 8 <--> 56 = (2)(2)(2)(7)

Okay, now onto the question at hand . . .

Target question: Is mn divisible by 40?

Since 40 = (2)(2)(2)(5), we can rephrase the target question as . . .

REPHRASED target question: Are there three 2's and one 5 hiding in the prime factorization of mn?

Statement 1: m is divisible by 10
There are several values of m and n that satisfy this condition. Here are two possible cases:
Case a: m = 10 and n = 8, in which case mn is divisible by 40
Case b: m = 10 and n = 3, in which case mn is not divisible by 40
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: n is divisible by 4
There are several values of m and n that satisfy this condition. Here are two possible cases:
Case a: m = 10 and n = 4, in which case mn is divisible by 40
Case b: m = 3 and n = 4, in which case mn is not divisible by 40
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1: Since 10 = (2)(5), statement 1 tells us that there is one 2 and one 5 hiding in the prime factorization of m.
Statement 2: Since 4 = (2)(2), statement 2 tells us that there are two 2's hiding in the prime factorization of n.
So, there must be three 2's and one 5 hiding in the prime factorization of mn

Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Cheers,
Brent

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by Brent@GMATPrepNow » Mon Nov 11, 2013 2:05 pm
dddanny2006 wrote:Hey Brent,

Can you explain this problem in terms of
Statement one tells us --2 x 5...

Statement two tells us --2 x 2...

I want to prove insufficiency based on the above concept.
Sure thing.

Statement 1: m is divisible by 10
Since 10 = (2)(5), statement 1 tells us that there is one 2 and one 5 hiding in the prime factorization of m.
So, we can be certain that there is at least one 2 and one 5 hiding in the prime factorization of mn.
Can we then conclude that there MUST be three 2's and one 5 hiding in the prime factorization of mn? No.
It COULD be the case that three 2's and one 5 are hiding in the prime factorization of mn, but it COULD be the case that three 2's and one 5 are not hiding in the prime factorization of mn

As such, statement 1 is NOT SUFFICIENT

Statement 2: n is divisible by 4
Since 4 = (2)(2), statement 2 tells us that there are two 2's hiding in the prime factorization of n.
So, we can be certain that there are two 2's hiding in the prime factorization of mn.
Can we then conclude that there MUST be three 2's and one 5 hiding in the prime factorization of mn? No.
It COULD be the case that three 2's and one 5 are hiding in the prime factorization of mn, but it COULD be the case that three 2's and one 5 are not hiding in the prime factorization of mn
As such, statement 2 is NOT SUFFICIENT

Does that help?

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com

Master | Next Rank: 500 Posts
Posts: 209
Joined: Thu Jan 12, 2012 12:59 pm
by dddanny2006 » Mon Nov 11, 2013 2:11 pm
Wow,brilliant.Thanks champ.Do you suggest testing values to be a better method than the prime factorization method?Which one's better and helpful in the longrun?Im not that comfortable with similar problems,whats the best way to master these?
Brent@GMATPrepNow wrote:
dddanny2006 wrote:Hey Brent,

Can you explain this problem in terms of
Statement one tells us --2 x 5...

Statement two tells us --2 x 2...

I want to prove insufficiency based on the above concept.
Sure thing.

Statement 1: m is divisible by 10
Since 10 = (2)(5), statement 1 tells us that there is one 2 and one 5 hiding in the prime factorization of m.
So, we can be certain that there is at least one 2 and one 5 hiding in the prime factorization of mn.
Can we then conclude that there MUST be three 2's and one 5 hiding in the prime factorization of mn? No.
It COULD be the case that three 2's and one 5 are hiding in the prime factorization of mn, but it COULD be the case that three 2's and one 5 are not hiding in the prime factorization of mn

As such, statement 1 is NOT SUFFICIENT

Statement 2: n is divisible by 4
Since 4 = (2)(2), statement 2 tells us that there are two 2's hiding in the prime factorization of n.
So, we can be certain that there are two 2's hiding in the prime factorization of mn.
Can we then conclude that there MUST be three 2's and one 5 hiding in the prime factorization of mn? No.
It COULD be the case that three 2's and one 5 are hiding in the prime factorization of mn, but it COULD be the case that three 2's and one 5 are not hiding in the prime factorization of mn
As such, statement 2 is NOT SUFFICIENT

Does that help?

Cheers,
Brent

Master | Next Rank: 500 Posts
Posts: 209
Joined: Thu Jan 12, 2012 12:59 pm
by dddanny2006 » Mon Nov 11, 2013 2:15 pm
@ Brent

Could you please help me out with this problem too,in the same method that you last solved the oter one for me.

Is m divisible by 8?
1.mn is divisible by 40
2.n is divisible by 5

Please solve it the same way you last did.
Thanks again.
Brent@GMATPrepNow wrote:
dddanny2006 wrote:Hey Brent,

Can you explain this problem in terms of
Statement one tells us --2 x 5...

Statement two tells us --2 x 2...

I want to prove insufficiency based on the above concept.
Sure thing.

Statement 1: m is divisible by 10
Since 10 = (2)(5), statement 1 tells us that there is one 2 and one 5 hiding in the prime factorization of m.
So, we can be certain that there is at least one 2 and one 5 hiding in the prime factorization of mn.
Can we then conclude that there MUST be three 2's and one 5 hiding in the prime factorization of mn? No.
It COULD be the case that three 2's and one 5 are hiding in the prime factorization of mn, but it COULD be the case that three 2's and one 5 are not hiding in the prime factorization of mn

As such, statement 1 is NOT SUFFICIENT

Statement 2: n is divisible by 4
Since 4 = (2)(2), statement 2 tells us that there are two 2's hiding in the prime factorization of n.
So, we can be certain that there are two 2's hiding in the prime factorization of mn.
Can we then conclude that there MUST be three 2's and one 5 hiding in the prime factorization of mn? No.
It COULD be the case that three 2's and one 5 are hiding in the prime factorization of mn, but it COULD be the case that three 2's and one 5 are not hiding in the prime factorization of mn
As such, statement 2 is NOT SUFFICIENT

Does that help?

Cheers,
Brent

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by Brent@GMATPrepNow » Wed Nov 13, 2013 9:00 am
dddanny2006 wrote:@ Brent

Could you please help me out with this problem too,in the same method that you last solved the other one for me.

Is m divisible by 8?

1) mn is divisible by 40
2) n is divisible by 5
Target question: Is m divisible by 8?

Since 8 = (2)(2)(2), we can rephrase the target question as follows:
REPHRASED target question: Are three 2's hiding in the prime factorization of 8?

Statement 1: mn is divisible by 40
Since 40 = (2)(2)(2)(5), this tells us that there are three 2's and one 5 hiding in the prime factorization of mn.
However, we can't be certain whether or not there are three 2's hiding in the prime factorization of m.
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Aside: we can also show that statement 2 is not sufficient by considering these two contradictory cases.
Case a: m = 8 and n = 5, in which case m is divisible by 8
Case b: m = 4 and n = 10, in which case m is not divisible by 8
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: n is divisible by 5
This tells us nothing about m
So, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Case a: m = 8 and n = 5, in which case m is divisible by 8
Case b: m = 4 and n = 10, in which case m is not divisible by 8
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com

Master | Next Rank: 500 Posts
Posts: 209
Joined: Thu Jan 12, 2012 12:59 pm
by dddanny2006 » Fri Nov 15, 2013 8:34 am
Hey Brent,

What if I change Statement 1 to mn is divisible by 8,everything else stays the same

Thanks
Dan
Brent@GMATPrepNow wrote:
dddanny2006 wrote:@ Brent

Could you please help me out with this problem too,in the same method that you last solved the other one for me.

Is m divisible by 8?

1) mn is divisible by 40
2) n is divisible by 5
Target question: Is m divisible by 8?

Since 8 = (2)(2)(2), we can rephrase the target question as follows:
REPHRASED target question: Are three 2's hiding in the prime factorization of 8?

Statement 1: mn is divisible by 40
Since 40 = (2)(2)(2)(5), this tells us that there are three 2's and one 5 hiding in the prime factorization of mn.
However, we can't be certain whether or not there are three 2's hiding in the prime factorization of m.
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Aside: we can also show that statement 2 is not sufficient by considering these two contradictory cases.
Case a: m = 8 and n = 5, in which case m is divisible by 8
Case b: m = 4 and n = 10, in which case m is not divisible by 8
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: n is divisible by 5
This tells us nothing about m
So, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Case a: m = 8 and n = 5, in which case m is divisible by 8
Case b: m = 4 and n = 10, in which case m is not divisible by 8
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Cheers,
Brent

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by Brent@GMATPrepNow » Fri Nov 15, 2013 8:52 am
dddanny2006 wrote:Hey Brent,

What if I change Statement 1 to mn is divisible by 8,everything else stays the same

Thanks
Dan
Hey Dan,

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com

Master | Next Rank: 500 Posts
Posts: 209
Joined: Thu Jan 12, 2012 12:59 pm
by dddanny2006 » Fri Nov 15, 2013 9:18 am
Cool Brent.

https://www.beatthegmat.com/trickyds-que ... tml#701946

Brent@GMATPrepNow wrote:
dddanny2006 wrote:Hey Brent,

What if I change Statement 1 to mn is divisible by 8,everything else stays the same

Thanks
Dan
Hey Dan,