79 I was wondering if you could give me a step by step solut

79.) I was wondering if you could give me a step by step solution to this problem so that I can better understand it. Thank you!

denotes z E Zn in our usual less cumbersome notation. This problem involves several moduli so I invoke the clumsy notation as to reduce ambiguity.

x=20 mod 37

So, x=37m+20 ,where m is an integer

x=4 mod 7

So,

37m+20=4 mod 7

(35+2)m+14+6=4 mod 7

2m+6=4 mod 7

2m=-2 mod 7

m=-1=6 mod 7

So, m=7k+6 , k is an integer

x=37m+20=37(7k+6)+20=259k+242

But, x=3 mod 2=1 mod 2

So, 259k+242=1 mod 2

259k=1 mod 2

Hence, k=1 mod 2

k=2n+1 , n is any integer

x=259k+242=259(2n+1)+242=516n+501

x=516n+501 , n is any integer