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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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New post 31 Mar 2017, 06:16
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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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New post 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.

Answer D.
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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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New post 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

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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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