If $6^{83} + 8^{83}$ is divided by $49,$ then the remainder is

35

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5

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1

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Solution

The correct option is A $35$
$6^{83} + 8^{83} = (1 + 7)^{83} + (7 - 1)^{83}$
$= (1 + 7)^{83} - (1 - 7)^{83}$
$= 2 [^{83} C_{1} \cdot 7 +^{83} C_{3} \cdot 7^{3} + \dots +^{83} C_{83} \cdot 7^{83}]$
$= (2 \cdot 7 \cdot 83) + 49 I,$ where $I$ is an integer.
$2 \times 7 \times 83 = 1162$
$\frac{1162}{49} = 23 \frac{35}{49}$
$∴$ Remainder is $35.$

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