Question

find the remainder when 7 to the power 21 plus 7 to the power 22 plus 7 to the power 23 plus 7 to the power 24 is divided by 25?

Answer 1

it is divisible by 25 so there is no remainder
normal way:
$\frac{7^{21}+7^{22}+7^{23}+7^{24}}{25}= \frac{223418345633313602800}{25}= 8936733825332544112$
smart way:
$\frac{7^{21}+7^{22}+7^{23}+7^{24}}{25}= \frac{7^{21}(1+7+49+343)}{25}= \frac{7^{21}(400)}{25}=4(7^{21})$