REMAINDER

23^17=(23-19)^17=4^17(mod 19) ,

15^17=(15-19)^17=(-4)^17=-(4^17)(mod 19) ,

4^17+(-4^17)=0(mod 19)

this is my solution.. but it is hard enough

• 23^(17) mod 19
• 4^(17) mod 19 // 4 is get from 23 - 19.
• 16^(8) . 4 mod 19
• -3^(8). 4 mod 19 // -3 is get from 16 - 19 = -3.
• 9^(4) . 4 mod 19
• 81^(2) . 4 mod 19
• 5^(2) . 4 mod 19

• 25 .4 mod 19 = 5;

• 15^(17) mod 19

• -4^(17) mod 19 // this is same like above.. so i skip it. to be..
• 25 . - 4 mod 19
• 6 . -4 mod 19 = -5

• 19 - 5 = 14.

• so 5 + 14 = 19, and 19 mod 19 is 0;

23^17+15^17 is divisible by 23+15=38 and 38 is also divisible by 19.

so, when 23^17+15^17 is divisivle by 19.

hence the remainder is 0 or zero.