[Math] Use the construction in the proof of the Chinese remainder theorem

Question

20. Use the construction in the proof of the Chinese remainder
    theorem to find all solutions to the system of congruences
    x ≡ 2 (mod 3), x ≡ 1 (mod 4), and x ≡ 3 (mod 5).

I am not sure what is the process of answering this question!?

Answer 1

Remainders :

r1 = 2

r2 = 1

r3 = 3

Multiply all the divisors as M:

M = 3*4*5 = 60

a1 = 60/3 = 20

a2 = 60/4 = 15

a3 = 60/5 = 12

NOW Inverse:

i1 = 20 mod 3 (inverse) = 2

i2 = 15 mod 4 (inverse) = 3

i3 = 12 mod 5 (inverse) = 3

Z = (i1.r1.a1)+(i2.r2.a2)+(i3.r3.a3)

Z = (2*2*20) + (3*1*15) + (3*3*12)

Z = 233

233 = x mod M

233 = x mod 60

Simply divide 233 by 60 and then the answer is the remainder:

x = 53. Answer