Question

Find the remainder when ${25}^{10}$ is divided by $576$.

Answer 1

${25}^{10}={(24+1)}^{10}\Rightarrow$ in the expression, total $11$ terms having precess of $24$ from $0$ to $10$

If power of $24\ge 2$ in a term, term divisible by $576$

Terms not divisible by $576$ are these having powers of $24$ as $0$ or $1$

$\therefore$ Terms are $:{}^{10}{C}_1\times {24}^1\times 1^9+{}^{10}{C}_2\times {24}^0\times 1^{10}$

$=10\times 24\times 1+1\times 1\times 1$

$=241$

$\therefore$ Remainder $=241$ Ans