29. Let $a, b$ be two natural numbers. If the product of $a$ and $b$ leaves a remainder of 1 when divided by 15, then $b$ is called the inverse of $a$ modulo 15. According to this definition, the sum of all inverses of 7 modulo 15 between 100 and 200 is ( )
(A) 1036
(B) 933
(C) 843
(D) 296
Answer: (A)

Solution: Let $7 x=15 q+1,7(x-2 q)-1=q=7 s-1$, where
$$
s=x-2 q . s=x-2 q, x=s+2 q=s+2(7 s-1)=15 s-2, s=1,2, \cdots \text {. }
$$

Between 100 and 200, $100<x=15 s-2<200,6<s<14$,
so there are 7 numbers that meet the requirement:
$$
103,118,133,148,163,178,193 \text {. }
$$

Their sum is 1036.