Q) What is the remainder when 5³⁸ is divided by 11?

A)
Remainders of exponential powers of 5 when divided by 11 come as 5, 3, 4, 9, 1 repetitively.
( Rem. 5 = 5
Rem. 25 = 3
Rem. 125 = 4
Rem. 625 = 9
Rem. 3125 = 1
And again Rem. 15625 = 5)

repetition occurs after every 5 terms....so 38th power will come as a third term of the sequence (7x5 + 3)
so the remainder is 4.

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