Question

Show that ( 19 93 - 13 99 ) is a POSITIVE INTEGER divisible by 162.

Answer 1

Both 19^93 and 13^99 are odd, so odd – odd is even and is divisible by 2, so it’s enough we need to show that 19^93 and 13^99 is divisible by (162 / 2) which is 81. 

Now, 19^93 (modulo 81) = (18 + 1)^93 (modulo 81) = 93(18) + 1 = 55 (modulo 81) 

Similarly, 13^99 (modulo 81) = (12 + 1)^99 (modulo 81) = 99(12) + 1 = 1189 (modulo 81) = 55 (modulo 81) 

Hence, 19^93 – 13^99 = 0 (modulo 81) 

Therefore, 19^93 and 13^99 is divisible by 162