Numbers : Remainder 105

find the remainder  3^5^7^9 divided by 41.

Soln :
3^4 = 81 = 41(2) - 1. So 3^8 will leave remainder 1 when divided by 41.
That means now we need to find the remainder of power of 3 i.e. 5^7^9 with 8.
Now we know that 5² = 25 = 8(3) + 1 i.e. 57^9 is of the form 52k + 1 and leave remainder 51 = 5 when divided by 8 and is of the form 8k + 5.So finally we need to find the remainder when 38k + 5 is divided by 41 or the remainder when 35 is divided by 41 which is -3 or 38.