Find the remainder when $7^{21} + 7^{22} + 7^{23} + 7^{24}$ is divided by $25$

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Solution

The correct option is A $0$
The remainder when $7^{1} \div 25 = 7 7^{2} \div 25 = 24 7^{3} \div 25 = 18 7^{4} \div 25 = 1 7^{5} \div 25 = 7$ and the cycle continues.
So, remainder when $7^{21} \div 25 = 7 7^{22} \div 25 = 24 7^{23} \div 25 = 18 7^{24} \div 25 = 1$ .
The sum of the remainders is $50$ .
$50$ divided by $25$ leaves a remainder of $0$ .

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