number theory - Least non negative residue

Find the least non negative residue of 17x18(mod35)

I was under the impression that you'd take 17 ≡ -18 and 18 ≡ -17 but obviously
(-17)(-18) is greater than 35 so this cannot be the answer

Using a calculator, compute the residue of 63433 x 723239(mod123433437)

I changed to the number theory course recently so have missed lectures on this and the lecture notes are very brief and rather unhelpful with solving these problems, any help would be greatly appreciated.

Answer

1. Note that $2\times 18 \equiv 36 \equiv 1 \pmod {35},\,$ so you get
   
   $17\times 18 \equiv 16\times 18 + 18 \equiv 8(2\times 18) + 18 \equiv 8 + 18 \equiv 26 \pmod {35}$




2. Compute $a = 63433\times 723239 = 45877219487\;$ and $a/123433437 \approx 371.68\;$ and therefore
   $a \equiv 45877219487-123433437\times 371 \equiv 83414360 \pmod {123433437}.$
   This is a special case of
   $a \bmod b = a -\lfloor a/b \rfloor b$