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If m and n are positive integers, and m + n and m/n are both even numb

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Bunuel
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If m and n are positive integers, and m + n and m/n are both even numb [#permalink] New post   28 Feb 2023, 12:30
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If m and n are positive integers, and m + n and m/n are both even numbers, which of the following cannot be an even integer ?

A. (m + n)/2

B. m/2

C. (n + 2)/2

D. (m + 4)/2

E. (m + 6)/2
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Re: If m and n are positive integers, and m + n and m/n are both even numb [#permalink] New post   09 Mar 2023, 08:13
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Bunuel wrote:
If m and n are positive integers, and m + n and m/n are both even numbers, which of the following cannot be an even integer ?
A. (m + n)/2
B. m/2
C. (n + 2)/2
D. (m + 4)/2
E. (m + 6)/2

Solution:

  • We are given \(m+n=even\)
    • This means m and n are either both even or both odd
  • It is also given that \(\frac{m}{n}=even\)
  • Combining both the info, we can infer that m and n are both even and m is even multiple of n
  • Let us assume \(n=2p\) and \(m=2q*2p=4pq\) where p and q are positive integers

  • Option A: (m + n)/2
    • \(\frac{2p+4pq}{2}=p+2pq=\) may be even

  • Option B: m/2
    • \(\frac{4pq}{2}=2pq=\) even

  • Option C: (n + 2)/2
    • \(\frac{2p+2}{2}=p+1=\) may be even

  • Option D: (m + 4)/2
    • \(\frac{4pq+4}{2}=2pq+2=\) even

  • Option E: (m + 6)/2
    • \(\frac{4pq+6}{2}=2pq+3=\) definitely not even

Hence the right answer is Option E
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Re: If m and n are positive integers, and m + n and m/n are both even numb [#permalink] New post   02 Mar 2023, 03:34
Bunuel wrote:
If m and n are positive integers, and m + n and m/n are both even numbers, which of the following cannot be an even integer ?

A. (m + n)/2

B. m/2

C. (n + 2)/2

D. (m + 4)/2

E. (m + 6)/2


Picking numbers worked best for me. Please note : Once we have tested an option we need not test that option again.

M\(=4\) and n\(=2 \)

B, C and D can be eliminated as all of these give even.

M\(=8\) and n\(=4\)

We can eliminate A

Ans E

Hope it helps.
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Re: If m and n are positive integers, and m + n and m/n are both even numb [#permalink] New post   09 Mar 2023, 08:34
Asked: If m and n are positive integers, and m + n and m/n are both even numbers, which of the following cannot be an even integer ?

m + n is an even number;
m & n both are even
or
m & n both are odd

m/n is an even number
m is an even number and is a multiple of 4.
n is an even number

A. (m + n)/2: even number

B. m/2 : even number

C. (n + 2)/2 = n/2 + 1: May be even or odd

D. (m + 4)/2 = m/2 + 2: even number

E. (m + 6)/2 = m/2 + 3: odd number

IMO E
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Re: If m and n are positive integers, and m + n and m/n are both even numb [#permalink] New post   27 Apr 2024, 02:58
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Re: If m and n are positive integers, and m + n and m/n are both even numb [#permalink]
27 Apr 2024, 02:58
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