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# If a, b and c are integers, is (a·b) a multiple of 18?

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Joined: 21 Oct 2013
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If a, b and c are integers, is (a·b) a multiple of 18?  [#permalink]

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27 Jun 2014, 12:55
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85% (hard)

Question Stats:

45% (01:51) correct 55% (02:15) wrong based on 133 sessions

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If a, b and c are integers, is (a·b) a multiple of 18?

(1) 2a = 3b
(2) 2b = 3c
Math Expert
Joined: 02 Sep 2009
Posts: 52385
Re: If a, b and c are integers, is (a·b) a multiple of 18?  [#permalink]

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27 Jun 2014, 13:04
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If a, b and c are integers, is (a·b) a multiple of 18?

Notice that we are told that all variables are integers.

(1) 2a = 3b --> a/b = 3/2. a is a multiple of 3 and b is a multiple of 2. ab must be a multiple of 6 but may or may not be a multiple of 18 (for example consider (a,b) = (3,2) and (a,b) = 9,6). Not sufficient.

(2) 2b = 3c --> b/c = 3/2. b is a multiple of 3. Not sufficient.

(1)+(2) b is a multiple of both 2 and 3, so it must be a multiple of 6 and since a is a multiple of 3, then ab must be a multiple of 6*3 = 18. Sufficient.

Hope it's clear.
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Re: If a, b and c are integers, is (a·b) a multiple of 18?  [#permalink]

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19 Aug 2014, 13:42
it is not mentioned that integers are different. case when all can be zero, hence i go for E
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Re: If a, b and c are integers, is (a·b) a multiple of 18?  [#permalink]

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19 Aug 2014, 13:47
1
satya14141 wrote:
it is not mentioned that integers are different. case when all can be zero, hence i go for E

Please check the OA (Official Answer) under the spoiler in the original post. It's C, not E.

Even if ab = 0, it's still divisible by 18: 0 is divisible by EVERY integer except 0 itself.

Check for more here: number-properties-tips-and-hints-174996.html

Hope this helps.
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Re: If a, b and c are integers, is (a·b) a multiple of 18?  [#permalink]

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25 Feb 2016, 19:06
tough one..now I see where I made the mistake..
2a=3b -> b is a multiple of 2, and a is a multiple of 3..
a*b=1.5b^2 -> b=6 -> yes, b=2 - no

2. 2b=3c
so b is a multiple of 3, and c is a multiple of 2. not sufficient.

1+2
b - multiple of both 2 and 3, so should be at least 6.
a is a multiple of 3, so should at least be 3.

now, 18 = 2*3^2.
a*b -> at least 2^1* 3^2.
so sufficient.
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Re: If a, b and c are integers, is (a·b) a multiple of 18?  [#permalink]

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13 Mar 2016, 22:48
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Re: If a, b and c are integers, is (a·b) a multiple of 18?  [#permalink]

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20 Apr 2018, 12:41
Top Contributor
1
goodyear2013 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.
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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Re: If a, b and c are integers, is (a·b) a multiple of 18? &nbs [#permalink] 20 Apr 2018, 12:41
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