Number Theory

We can write a handy equation to solve this problem. 

First, let's set a variable. Let's set x as a positive integer. 

Any number congruent to 7 modulo 29 can be written in the form

\(29x+7\) since it is 7 greater than a factor of 29. 

 

thus, we can write the inequality

\(29x+7\leq 1000\)

 

Setting the two to equal and solving for x, we have

\(29x+7=1000\\ 29x=993\\ x \approx 34.24\)

 

Rounding down, we find x is 34. 

So our answer is 34. 

 

Thanks! :)