Q) Find the remainder when 7^23 is divided by 25 .

A) 7^23 = (7^22) . 7
= [ (7^2)^11] . 7
= [(50 - 1)^11] .7
Now In expansion of (50 - 1)^11 , all terms except the last one will be divisible by 25 since they contain an exponent of 50 in them.
Last term = 11C11 * (-1)
So we are left with
(-1)(7)
= -7
so 25 - 7 = 18 which is the remainder

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