Maths Problem On Congruences?

Question

a) Find the remainder when 27^1001 is divided by 13

b) Find the remainder when 38^101 is divided by 13

c) Hence show that 70x27^1001+31x38^101 is divisible by 13

Please use congruences and show all your working.

THANKS

Answer 1

27 = 1 mod 13 as 27 = 2*13 +1

so 27^1001 = 1^ 1001 = 1  mod 13

38 = - 1 mod 13  

so 38^ 101 = (-1) ^101 = -1 mod 13

so

70x27^1001+31x38^101 mod 13

= 70*1 - 13 *1 = 39 mod 13 = 0

so 70x27^1001+31x38^101 is divisible by 13