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Finding Remainder of Numbers of the Form a^b

If 883+683 is...

Question

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

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Solution

$8^{83} + 6^{83} = (1 + 7)^{83} - (1 - 7)^{83} = 2 [^{83} C_{1} 7 +^{83} C_{3} 7^{3} + \dots +^{83} C_{83} 7^{83}] = 2 \times^{83} C_{1} 7 + 49 k, k \in Z = (2 \times 83 \times 7) + 49 k = 166 \times 7 + 49 k = 161 \times 7 + 49 k + 5 \times 7 = 49 (k + 23) + 35$

Therefore, the remainder is $35$ .

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Finding Remainder of Numbers of the Form a^b

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