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A Party Trick

If someone tells me 2, 2 and 3 are the remainders when she divides her age by 3, 5 and 7 respectively then I can work out her age.

Solution:

Let x = 2, y = 2, z = 3 and a = the age of the girl. Then she used this formula:

a = (70x + 21y + 15z)mod n

= (120 + 42 + 45)mod (3 x 5 x 7)

= 227mod105

= 17 years old

I have no idea of how this works. Where does this formula come from? There must be a logical way to explain this right? And are there any other ways of solving this kind of problem?