Solve the following systems of simultaneous congruences Sol

. Solve the following systems of simultaneous congruences:

Solve the following systems of simultaneous congruences: { 2x 5 mod 27 x 5 mod 84 4x 57 mod 63

2x=5 mod 27

MUltiply both sides by 14 gives

28x=70 mod 27

x=16 mod 27

x=5 mod 84

4x=57 mod 63

MUltiplying both sides by 16 gives

64x=57*16=-6*16=-96=-33=30

x=30 mod 63

x=16 mod 27

x=5 mod 84

x=30 mod 63

But 63=7*9

So,

x=30 mod 63

So,

x=3 mod 9

x=16 mod 27

Hence, x=16 mod 9

Hence, x=7 mod 9

But we just found,

x=2 mod 9

SO no solution exists.