JEE Main 2024 (Online) 29th January Evening ShiftQuestion 1
Remainder when $64^{32^{32}}$ is divided by $9$ is equal to ______.
Binomial Theorem · Remainder and Divisibility
Answer$1$
Solution
Concept. $64$ is one more than a multiple of $9$. Formulas used. (1) $(1+9m)^N=1+(\text{multiple of }9)$ by the binomial theoremStep 1 - Write $64=1+63$. $63=9\cdot7$, so by (1) $$64^{32^{32}}=(1+63)^{32^{32}}=1+(\text{multiple of }9).$$ Step 2 - Remainder. The remainder is $1$. Answer: 1