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# For positive integers m and n, is 7+72+73+….+7mn divisible by 14? 1) n

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For positive integers m and n, is 7+72+73+….+7mn divisible by 14? 1) n  [#permalink]

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31 Mar 2017, 06:16
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75% (hard)

Question Stats:

48% (02:20) correct 52% (02:16) wrong based on 50 sessions

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For positive integers m and n, is $$7+7^2+7^3+….+7^{mn}$$ divisible by 14?

1) $$n=even$$

2) $$m = \frac{a}{b}$$ , where one of a and b is an odd and the other is an even

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Re: For positive integers m and n, is 7+72+73+….+7mn divisible by 14? 1) n  [#permalink]

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31 Mar 2017, 08:09
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ziyuen wrote:
For positive integers m and n, is $$7+7^2+7^3+….+7^{mn}$$ divisible by 14?

1) $$n=even$$

2) $$m = \frac{a}{b}$$ , where one of a and b is an odd and the other is an even

Hi

Good question +1

$$7+7^2+7^3+….+7^{mn}$$ = $$7*(1 + 7 + 7^2 + 7^3 + .... + 7^{mn - 1})$$

The problem is whether the expression $$(1 + 7 + 7^2 + 7^3 + .... + 7^{mn - 1})$$ is even or odd.

1) $$n=even$$

If n is even then $$mn - 1 = odd$$ and we have = 1 + sum of odd number of odd elements (please excuse me for tautology) because $$7^{any power}$$ will be odd.

$$1 + odd = even$$. Sufficient.

2) $$m = \frac{a}{b}$$ , where one of a and b is an odd and the other is an even

So if $$a=odd$$ then $$b=even$$ and vice versa. But if we want $$m$$ to be an integer, first case will be impossible ($$\frac{odd}{even} =/= integer$$).

And we are left with $$m*odd = even$$ hence $$m$$ is even.

As in previous cae $$mn - 1$$ will be odd. Sufficient.

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Re: For positive integers m and n, is 7+72+73+….+7mn divisible by 14? 1) n  [#permalink]

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01 Apr 2017, 09:35
ziyuen wrote:
For positive integers m and n, is $$7+7^2+7^3+….+7^{mn}$$ divisible by 14?

1) $$n=even$$

2) $$m = \frac{a}{b}$$ , where one of a and b is an odd and the other is an even

7(1+7+7^2+7^3+….+7^mn-1)
For the above to be divisible by 14. The expression in bracket has to be divisible by 2. So the expression needs to be even. 1 is not divisible by 2, but 1+7 is. So the power of 7 has to be odd.
Therefore question boils down to....is mn-1=odd?
For mn-1 to be odd, mn should be even, for mn to be even either of them or both of needs to be even.

1) n= even; Suff
2)m=a/b. For this to be an integer m has to be even; Suff

Ans:D
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Re: For positive integers m and n, is 7+72+73+….+7mn divisible by 14? 1) n &nbs [#permalink] 01 Apr 2017, 09:35
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