Find the remainder when 1740 is divided by 27 using congruen

Find the remainder when 17^40 is divided by 27 using congruences/modular arithmetic

Solution

17 = -10 mod 27

=>

17^2 = 19 mod 27

17^4 = 10 mod 27

17^8 = 19 mod 27

17^16 = 10 mod27

17^32 = 19 mod 27

17^40 = 17^32 * 17^8 mod 27

= (19 + 19) mod 27

= 38 mod 27

= 11 mod 27

hence the remainder is 11