Question

Remainder obtained when $4^{100}$ is divided by $127$ is

• A. $4$
• B. $8$
• C. $16$
• D. $32$

Answer 1

The correct option is C $16$

$4^{100}={{(2}^2)}^{100}=2^{200}$

$=2^{196}\times 2^4$

$=2^{7\times 28}\times 2^4$

$=16\times {{(2}^7)}^{28}$

$=16\times {\left(128\right)}^{28}$

$=16\times {(127+1)}^{28}$

$=16\times [{}^{127}{C}_0{\left(127\right)}^{28}{-}^{127}{C}_1{\left(127\right)}^{27}+....+1]$

$=16\times (127\lambda +1);\lambda \in I$

$=16\times 127\lambda +16$

Therefore, when $4^{100}$ is divided by $127,$ the quotient will be $16\lambda$ & the remainder is $16.$

Hence, the answer is $16.$