Question

If $\lambda $ is the remainder when $2^{2021}$ is divided by $17,$ then the value of $\lambda $ must be equal to

• A. $3$
• B. $7$
• C. $13$
• D. $15$

Answer 1

Answer: (D)

$2^{2021}=2.2^{2020}=2.\left(2^{4}\right)^{505}=2\left(16\right)^{505}=2\left(17 - 1\right)^{505}$
$=2\left\{1 7^{505} - ^{505} C_{1} 1 7^{504} + ^{505} C_{2} 1 7^{503} . . . . . . . . . . . . + ^{505} C_{504} 1 7^{1} - ^{505} C_{505} 1\right\}$
$=2\left\{17 k - 1\right\}$
$=17\cdot 2k-2=17\cdot 2k-17+15$
$=17\left(2 k - 1\right)+15$