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## Divisible

tagged by: Brent@GMATPrepNow

This topic has 2 expert replies and 2 member replies
akhilsuhag Master | Next Rank: 500 Posts
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#### Divisible

Wed Nov 19, 2014 12:39 pm
If a, b and c are integers, is (aÂ·b) a multiple of 18?

(1) 2a=3b

(2) 2b=3c

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### GMAT/MBA Expert

Rich.C@EMPOWERgmat.com Elite Legendary Member
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Wed Nov 19, 2014 1:03 pm
Hi akhilsuhag,

This question is perfect for TESTing VALUES.

We're told that A, B and C are INTEGERS. We're asked if (A)(B) is a multiple of 18? This is a YES/NO question.

Fact 1: 2A = 3B

Ifâ€¦
A = 3
B = 2
AB = 6 and the answer to the question is NO

A = 9
B = 6
AB = 54 and the answer to the question is YES
Fact 1 is INSUFFICIENT

Fact 2: 2B = 3C

We know NOTHING about A, so this Fact is likely to be INSUFFICIENT, but here's how you can prove it:

Ifâ€¦
B = 3
C = 2
A = 1
AB = 3 and the answer to the question is NO

B = 3
C = 2
A = 6
AB = 18 and the answer to the question is YES
Fact 2 is INSUFFICIENT

Combined, we knowâ€¦
2A = 3B
2B = 3C

This tells us that B MUST be a multiple of 2 (from Fact 1) AND a multiple of 3 (From Fact 2), so B MUST be a multiple of 6â€¦By extension, this means that A MUST be a multiple of 3. TESTing VALUES proves it:

Ifâ€¦.
B = 6
A = 9
AB = 54 and the answer to the question is YES

B = 12
A = 18
AB = (18)(12) and the answer to the question is YES
Combined, SUFFICIENT

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

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Mathsbuddy Master | Next Rank: 500 Posts
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Thu Nov 20, 2014 5:28 am
If ab is a multiple of 18, then 18 is a factor of ab.
To be such, then ab = 2 x 3 x 3 x n, where n is a non-negative integer

Using Statement 1 we know a = 3b/2, which does not necessarily satisfy ab = 3b^2/2 = 2 x 3 x 3 x n
In fact, the factor of 3 is guaranteed only once. NOT SUFFICIENT

Using Statement 2 we know b = 3c/2, which does not necessarily satisfy ab = 3ac/2 = 2 x 3 x 3 x n
In fact, the factor of 3 is guaranteed only once. NOT SUFFICIENT

Combined: (2) ab = 3b/2 x 3c/2 = 9bc/4. Three is 2 of the factors here. In addition, the fact that a = 3b/2 means that for a to be an integer, then 3b must be a multiple of 2. Hence factors 2, 3 and 3 are guaranteed. This suggests: SUFFICIENT

However, do not ignore what happens if a and/or b equal zero (because zero is a multiple of all numbers). Quick test: if a and or b = 0, then ab = 0. Also SUFFICIENT.

### GMAT/MBA Expert

Brent@GMATPrepNow GMAT Instructor
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Sat Nov 22, 2014 7:13 am
akhilsuhag wrote:
If a, b and c are integers, is ab a multiple of 18?

(1) 2a = 3b
(2) 2b = 3c
Target question: Is ab a multiple of 18?

Given: a, b and c are INTEGERS

Statement 1: 2a = 3b
This statement doesn't FEEL sufficient, so I'm going to test some values.

Aside: For more on this idea of plugging in values when a statement doesn't feel sufficient, you can read my article: http://www.beatthegmat.com/mba/2013/10/25/data-sufficiency-when-to-plug-in-values

There are several values of a and b that satisfy this condition. Here are two:
Case a: a = 3 and b = 2, in which case ab = 6, and 6 is NOT a multiple of 18
Case b: a = 9 and b = 6, in which case ab = 54, and 54 IS a multiple of 18
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: 2b = 3c
There's no information about the variable a, so a can have ANY value. So, this statement SEEMS/FEELS insufficient. Let's test some values.
There are several values of a, b and c that satisfy this condition (keeping in mind that variable a can have ANY value). Here are two possible cases:
Case a: a = 1, b = 3 and c = 2, in which case ab = 3, and 3 is NOT a multiple of 18
Case b: a = 6, b = 3 and c = 2, in which case ab = 18, and 18 IS a multiple of 18
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that 2a = 3b
Divide both sides by 2 to get: a = 3b/2
Since a is an INTEGER, we know that 3b/2 is an INTEGER
If 3b/2 is an INTEGER, then b must be divisible by 2

Statement 2 tells us that 2b = 3c
Divide both sides by 3 to get: c = 2b/3
Rewrite as c = (2/3)b
Let's also take the statement 1 equation (2a = 3b) and divide both sides by 3 to get: b = 2a/3
Now take c = (2/3)b and replace b with 2a/3
We get: c = (2/3)(2a/3)
Simplify to get: c = 4a/9
Since c is an INTEGER, we know that 4a/9 is an INTEGER
If 4a/9 is an INTEGER then a must be divisible by 9

We now know that b must be divisible by 2 and a must be divisible by 9.
So, we can conclude that ab is divisible by (2)(9)
In other words, ab is a multiple of 18
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Cheers,
Brent

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Gurpreet singh Senior | Next Rank: 100 Posts
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Sun Jun 19, 2016 10:29 pm
Having knowledge is one thing but putting it in a language which can be understood by all is another skill.
I was lost in this question
Brilliant explanation Brent.
Regards

Brent@GMATPrepNow wrote:
akhilsuhag wrote:
If a, b and c are integers, is ab a multiple of 18?

(1) 2a = 3b
(2) 2b = 3c
Target question: Is ab a multiple of 18?

Given: a, b and c are INTEGERS

Statement 1: 2a = 3b
This statement doesn't FEEL sufficient, so I'm going to test some values.

Aside: For more on this idea of plugging in values when a statement doesn't feel sufficient, you can read my article: http://www.beatthegmat.com/mba/2013/10/25/data-sufficiency-when-to-plug-in-values

There are several values of a and b that satisfy this condition. Here are two:
Case a: a = 3 and b = 2, in which case ab = 6, and 6 is NOT a multiple of 18
Case b: a = 9 and b = 6, in which case ab = 54, and 54 IS a multiple of 18
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: 2b = 3c
There's no information about the variable a, so a can have ANY value. So, this statement SEEMS/FEELS insufficient. Let's test some values.
There are several values of a, b and c that satisfy this condition (keeping in mind that variable a can have ANY value). Here are two possible cases:
Case a: a = 1, b = 3 and c = 2, in which case ab = 3, and 3 is NOT a multiple of 18
Case b: a = 6, b = 3 and c = 2, in which case ab = 18, and 18 IS a multiple of 18
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that 2a = 3b
Divide both sides by 2 to get: a = 3b/2
Since a is an INTEGER, we know that 3b/2 is an INTEGER
If 3b/2 is an INTEGER, then b must be divisible by 2

Statement 2 tells us that 2b = 3c
Divide both sides by 3 to get: c = 2b/3
Rewrite as c = (2/3)b
Let's also take the statement 1 equation (2a = 3b) and divide both sides by 3 to get: b = 2a/3
Now take c = (2/3)b and replace b with 2a/3
We get: c = (2/3)(2a/3)
Simplify to get: c = 4a/9
Since c is an INTEGER, we know that 4a/9 is an INTEGER
If 4a/9 is an INTEGER then a must be divisible by 9

We now know that b must be divisible by 2 and a must be divisible by 9.
So, we can conclude that ab is divisible by (2)(9)
In other words, ab is a multiple of 18
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Cheers,
Brent

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