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

<br /> x = 31k_1 + 10 = 73k_2 + 35 = 111k_3+29, \tex{ then?}<br />

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,
<br /> x = 111\times73\times31\times u_1<br /> +73\times31\times u_2<br /> + 31\times k_3 + 10<br />
But it seems to be not so helpful.
Any ideas? thanks in advance.

 

[HallsofIvy]

This uses the "Chinese remainder theorem".