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If m and n are positive integers, is |m - n| + |2m - n| + |2n - m|

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Bunuel
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If m and n are positive integers, is |m - n| + |2m - n| + |2n - m| [#permalink] New post   12 Oct 2021, 01:30
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77% (01:58) correct 23% (02:39) wrong based on 130 sessions
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If m and n are positive integers, is \(|m - n| + |2m - n| + |2n - m|\) divisible by 4 ?

(1) \(2m > n\)
(2) \(2n > m\)
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E
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Re: If m and n are positive integers, is |m - n| + |2m - n| + |2n - m| [#permalink] New post   12 Oct 2021, 04:34
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|m - n| + |2m - n| + |2n - m|

1) 2m > n
Let m = n = 1
|m - n| + |2m - n| + |2n - m|
|1-1|+|2-1|+|2-1|
2 (2 is not divisible by 4)

Let M = 3 and N = 4
|m - n| + |2m - n| + |2n - m|
|3 - 4| + |2*3 - 4| + |2*4 - 3|
1+2+5
8 is divisible by 4

Not sufficient

2) 2N > M
Let m = n = 1
|m - n| + |2m - n| + |2n - m|
|1-1|+|2-1|+|2-1|
2 (2 is not divisible by 4)

Let M = 3 and N = 4
|m - n| + |2m - n| + |2n - m|
|3 - 4| + |2*3 - 4| + |2*4 - 3|
1+2+5
8 is divisible by 4

Not sufficient

Combining both,
Let M = 3 and N = 4
|m - n| + |2m - n| + |2n - m|
|3 - 4| + |2*3 - 4| + |2*4 - 3|
1+2+5
8 is divisible by 4

Let M = 3 and N = 2
|m - n| + |2m - n| + |2n - m|
|3 - 2| + |2*3 - 2| + |2*2 - 3|
1+4+1
6 is not divisible by 4

Answer is E

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Re: If m and n are positive integers, is |m - n| + |2m - n| + |2n - m| [#permalink] New post   05 Nov 2022, 09:07
Bunuel wrote:
If m and n are positive integers, is \(|m - n| + |2m - n| + |2n - m|\) divisible by 4 ?

(1) \(2m > n\)
(2) \(2n > m\)


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The way I solved is - I can quickly see that each of the statements alone cannot be possibly sufficient. On combining both the statements I know that the second and third modulus are positive, but I still don't know whether the first one is positive or not. (since m>=n or m<=n). Either way, I get either 2m or 2n, which may or may not be divisible by 4, depending on whether m and n are even or not. So even both statements combined are not sufficient, and the answer is E.
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Re: If m and n are positive integers, is |m - n| + |2m - n| + |2n - m| [#permalink]
05 Nov 2022, 09:07
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