Find x Given Remainders: Number Theory Problem and Solution

[ismaili]

Homework Statement

A given number $$x$$, if divided by 31, the remainder is 10, if divided by 73, the remainder is 35, if divided by 111, the remainder is 29. Then, what's the number $$x$$?

Homework Equations

$$x = 31k_1 + 10 = 73k_2 + 35 = 111k_3+29, \tex{ then?}$$

The Attempt at a Solution

I roughly remember this is a famous problem in high school mathematics, but I can't remember the way to solve this type of problems. The number of unknowns seem to be larger than the number of equations. I tried to write these equations in a way like,
$$x = 111\times73\times31\times u_1<br /> +73\times31\times u_2<br /> + 31\times k_3 + 10$$
But it seems to be not so helpful.
Any ideas? thanks in advance.

 

[HallsofIvy]

This uses the "Chinese remainder theorem".