Hi Sai,

One way to start is to write 63¹⁵ as (60+3)¹⁵, then if you know about expanding an expression like this using the binomial theorem, you can see that all terms in the expansion except the last clearly are multiples of 100 as they contain powers of 60, and so you only need to focus on the 10's digit in 3¹⁵ (you could just grind this out). Of course you must do 15⁶³ also, which would be hard to grind out, but you can see that the last two digits of powers of 15 quickly cycle through 25,75,25,75,... so that you can find the tens digit of this easily enough.

Penny