Question

The remainder when,
${10}^{10}\cdot ({10}^{10}+1)({10}^{10}+2)$ is divided by $6$ is ______.

• A. $2$
• B. $4$
• C. $0$
• D. $6$

Answer 1

Answer: (C)

$0$

${10}^{10}$ when divided by $6$ will give a remainder equal to $4^{10}$

Next, $4^{10}$ can be written as $(4^2{)}^5$ and so the remainder comes out to be $(16-12{)}^5$ i.e. $4^5$

$4^5$ can be written as $2^{10}$ or ${32}^2$

This when divided by $6$ would yield a remainder of $2^2$ i.e. $4$

Thus, the remainder when ${10}^{10}.({10}^{10}+1)({10}^{10}+2)$ is divided by $6$ becomes $4\times 5\times 6$

$120$ when divided by $6$ leaves remainder zero, implying that the remainder is zero.