# Proofs and numbers

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Imagine you are an old Chinese scholar teaching Proof and Numbersto Chinese princes. When you group
your pupils in rows of five, two are left over. If you group them in eights, only one is left over. So you
try it with rows of nine, but now you are left with seven pupils. What do you think would happen if
you grouped your pupils in rows of eleven? And can you guess how many pupils you have?

• Proofs and numbers - ,

You could do it by brute force
conditon 1: remainder of 2 when divided by 5 ...
7 12 17 22 27 32 ... 82 87 92 97 102 .....
condition 2: remainder of 1 when divided by 8
9 17 25 33 41 .... 89 97 105 ...
condition 3: remainder of 7 when divided by 9
16 25 34 .... 88 97 106 ...

ahhh, looks like 97 is in all three cases
So there could be 97 students
If grouped in rows of 11 , the remainder would be 9

A more rigorous method is to use the
"Chinese Remainder Theorem"
It is a rather complicated method and you must know how to manipulate expressions such as
a mod(b)
e.g. 47 = 2mod9

• Proofs and numbers - ,

If you want to know more about the Chinese Remainder Theorem , here is a good YouTube for it