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# If m and n are positive integers, is 36+36+m+n divisible by 4? 1) m

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If m and n are positive integers, is 36+36+m+n divisible by 4? 1) m  [#permalink]

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29 Mar 2017, 22:20
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Difficulty:

95% (hard)

Question Stats:

41% (02:30) correct 59% (02:35) wrong based on 81 sessions

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If m and n are positive integers, is $$3^6+3^{6+m+n}$$ divisible by 4?

1) $$m=3n+1$$
2) $$m+3n$$ is an odd number

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If m and n are positive integers, is 36+36+m+n divisible by 4? 1) m  [#permalink]

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29 Mar 2017, 23:21
1
ziyuen wrote:
If m and n are positive integers, is $$3^6+3^{6+m+n}$$ divisible by 4?

1) $$m=3n+1$$
2) $$m+3n$$ is an odd number

$$3^6+3^{6+m+n}$$ = $$3^6(1+3^{m+n})$$

the question is $$3^6+3^{6+m+n}$$ divisible by 4 reduces to is $$3^6(1+3^{m+n})$$ divisible by 4

since $$3^6$$ is not divisible by 4 we have to find if $$(1+3^{m+n})$$ is divisible by 4

St I
m=3n+1

$$(1+3^{m+n})$$ = $$(1+3^{3n+1+n})$$ = $$(1+3^{4n+1})$$

so for all the positive integer values of n, 4n+1 is an odd number and $$1+3^{Odd-Number}$$ is always divisible by 4 -----------Sufficient

St II
m+3n is an Odd number

which means m+n is also Odd number and $$1+3^{Odd-Number}$$ is always divisible by 4 ----------Sufficient

Hence option D is correct
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Re: If m and n are positive integers, is 36+36+m+n divisible by 4? 1) m  [#permalink]

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24 Aug 2017, 10:20
$$3^{6}+3^{6+m+n}$$ = $$3^{6}$$(1+$$3^{m+n}$$)
Thus if m+n is whether even or odd can give us a unique answer.
Statement 1:
m=3n+1
m-3n = 1 (odd), only possible when one of either m or n is odd and other is even. Which implies - m+n = odd.
Sufficient
Statement 2:
m+3n is odd,only possible when one of either m or n is odd and other is even. Which implies - m+n = odd.
Sufficient.
Thus option D
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Re: If m and n are positive integers, is 36+36+m+n divisible by 4? 1) m   [#permalink] 24 Aug 2017, 10:20
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