By using number theory and residue systems Show that 7 divid

By using number theory and residue systems:

Show that 7 divides 1941^1963 + 1963^1991

Solution

1941=2 mod 7

1963=3 mod 7

2^3=8=1 mod 7

1963=654*3+1

So,
1941^1963=2^{3*654+1}=(2^3}^654*2 mod 7=1^654*2 mod 7=2 mod 7

By Fermat\'s Little Theorem

3^6=1 mod 7

1991=1986+5=331*6+5

So,

1963^1991=3^{6*331+5}=(3^6)^331*3^5=3^5 mod 7=243=5 mod 7

HEnce

1941^1963+1963^1991=2+5=7 mod 7

Hence proved