Use Eulers theorem to show that 511032n 9 7 for any intege

Use Euler\'s theorem to show that 51|(10^32n + 9 - 7) for any integer n greaterthanorequalto 0.

Solution

We have, (51) = (3*17) =(3)*(17) = (3-1) * (17-1)=2*16 = 32
and gcd(10, 51) = 1, Therefore, by Euler\'s Theorem we get, 10^32 = 1 (mod 51).
Now,
10^(32n+9) - 7

=10^(32n)*10^9-7
= (10^32)^n * 10^9 - 7
= 1^n * ((10^2)^4 * 10) - 7
= 1 * ((-2)^4 * 10) - 7
= 160 - 7
= 153
= 0 (mod 51).

That means 51 | (10^(32n+9) - 7).