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

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Bunuel
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If a, b and c are integers, is (a·b) a multiple of 18? [#permalink] New post   12 Mar 2021, 01:15
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If a, b and c are integers, is (a·b) a multiple of 18?

(1) 2a = 3b
(2) 2b = 3c

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Bunuel
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Re: If a, b and c are integers, is (a·b) a multiple of 18? [#permalink] New post   15 Mar 2021, 00:30
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Bunuel wrote:
If a, b and c are integers, is (a·b) a multiple of 18?

(1) 2a = 3b
(2) 2b = 3c

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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.

Answer: C.

Hope it's clear.
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If a, b and c are integers, is (a·b) a multiple of 18? [#permalink] New post   12 Mar 2021, 07:54
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imo C
If a, b and c are integers, is (a·b) a multiple of 18?

(1) 2a = 3b
b= 2a/3
if a = 3 , b = 2 => ab = 6
if a= 18 , b= 12 => ab = 18 (12) .. Multiple of 18 ...
Not sufficient

(2) 2b = 3c
No info about a so
not sufficient ..

no combining 1 and 2 ..

b= 2a/3
c = 2b/3
since minimum of b has to be 6 (in form of 6n , n 1,2,3 ...) to satisfy a b c are integer ..
if b = 6 , c = 4 , a = 9 ... ab = 18 (something) .. multiple of 18 ...
.. Sufficient
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Re: If a, b and c are integers, is (a·b) a multiple of 18? [#permalink] New post   12 Mar 2021, 21:54
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If a, b and c are integers, is (a·b) a multiple of 18?

Stat 1: 2a = 3b
It means, a = multiple of 3 and b= multiple of 2. Now, (a·b) can be multiple of 6, which means either multiple of 18 or not. Not sufficient

Stat 2: 2b = 3c
It means, b = multiple of 3 and c= multiple of 2. But we don't know anything about a. Not sufficient

Combining both, a = multiple of 3 and b= multiple of 2 & 3. Now, (a·b) can be multiple of 18. Sufficient

So, I think C. :)
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Re: If a, b and c are integers, is (a·b) a multiple of 18? [#permalink] New post   14 Mar 2021, 00:44
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If a, b and c are integers, is (a·b) a multiple of 18?

(1) 2a = 3b
a = 3b/2

Put b = 2, a = 3 and a.b = 6, which is not a multiple of 18

Put b = 6, a = 9 and a.b = 54, multiple of 18
Not sufficient

(2) 2b = 3c
b = 3c/2

Put c = 2, then b = 3 and put a = 1, a.b = 3

Put c = 2, then b = 3 and put a = 6, a.b = 18

Not sufficient

(1) + (2)

2b = 3c
b = 3c/2

2a = 3b
2a = 3(3c/2)
a = 9c/4

Since a, b, and C are integers. Minimum value of c = 4

For c = 4, a = 9, b = 6

a.b = 54

So, For any value of c, a is always a multiple of 9 and b is always a multiple if 6. Thus, a.b is always a multiple of 18

Sufficient

Option C

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Re: If a, b and c are integers, is (a·b) a multiple of 18? [#permalink] New post   16 Apr 2023, 06:53
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Re: If a, b and c are integers, is (a·b) a multiple of 18? [#permalink]
16 Apr 2023, 06:53
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