Solve the system of congruences Using Chinese Remainder Ther

Solve the system of congruences Using Chinese Remainder Theroram.

Exercise 40, page: 155 from AN INTRODUCTION TO Number Theory with Cryptography By Krafts and Washington.

x=2 mod 11

So, x=11m+2, m is any integer

4x=3 mod 7

4(11m+2)=3 mod 7

44m+8=3 mod 7

44m=-5 mod 7

44m=2 mod 7

2m =2 mod 7

m=1 mod 7

x=11m+2 =11(7n+1)+2=77n+13

3x=2 mod 5

3(77n+13)=2 mod 5

3(2n+3)=2 mod 5

6n+9=2 mod 5

n=-7 =3 mod 5

So,

x=77n+13=77(5k+3)+13=385k+234