Chinese remainder theorem

One of the most famous technique that comes from china is know as the Chinese remainder theorem. The theorem is used to solve problems dealing with system of linear congruence and its earliest uses is dated back in the book Mathematical Classic Master sun.

Example:
"We have things of which we do not know the number: if we count them by threes, the remainder is 2; if we count them by fives, the remainder is 3; if we count them by sevens , the remainder is 2. How many things are there?"

This is translated into
N= 3x+2
N=5y+3
N=7z+2

or for those with a background in number theory
N== 2(mod 3)
N== 3(mod 5)
N== 2 (mod 7)

where == stands for congruence. For simpler purpose, we will use the earlier notation. So,
1) 3*7=21
     21 /5 give you a remainder 1
    so 3*21= 63

2) 3*5= 15
    15/ 7 gives you a remainder 1
  so 2* 15= 30

3) 5*7 = 35
35*2= 70   we multiply again because 35/ 3 gives you a remainder 2. We want a remainder of 1
70/3 gives you a remainder 1
70*2 = 140


Then adding 63 + 30 + 140 = 233
so, 233-210 = 23                             210= 2(3*5*7)
thus N= 23

When plunging in our answer we see that it satisfies the system of congruences.
Did you notice any patterns during the steps? 
In number theory terms:

• We multiplied two mods together and then divided by the third and we did that for each of the mod
• When we divide the product by the third mod and got a remainder of 1 we multiplied the product by the remainder of that mod

Unanswered Question:

1. Was the Chinese remainder theorem very applicable? I find it hard to find a situation where the objects are unable to be counted

Evaluation:
Interesting: 7
Complexity: 6
Quality: 8