Question

How many incongruent solutions does

x³ ≡ 1 have modulo 59? mod 61?

x³ ≡ 8 mod 59 and 61? How about the congruences y⁵ ≡ 1 and y⁵ ≡ 32 mod 59 and 61?

Explain the results you get and why they are the case.

Answer 1

x3 1 (mod 59) implies that x 1(mod 59) and mod 61.

x3 8 (mod 61) implies that x 2(mod 61) and mod 59.

y5 1 (mod 59) implies that x 1(mod 59) and mod 61.

y532 (mod 59) implies that x2 (mod 59) and mod 61.

All will have unique incongruent solution as by CRT we have d= gcd (a,n)=1.

They are the case because 59 and 61 are primes. and d= (1,59)= 1.