Question

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

• A. 35
• B. 5
• C. 1
• D. 0

Answer 1

Answer: (A)

$(1+7)^{83}+(7-1)^{83}=(1+7)^{83}-(1-7)^{83}$
$=2\left[{ }^{83} C _{1} \cdot 7+{ }^{83} C _{3} \cdot 7^{3}+\cdots+{ }^{83} C _{83} \cdot 7^{83}\right]=(2 \cdot 7 \cdot 83)+49 I$
where $I$ is an integer
Now, $14 \times 83=1162$
$\frac{1162}{49}=23 \frac{35}{49}$
Therefore, remainder is $35 .$