Question

Find the integer a such that a ≡ 08(mod23) and −22 ≤ a ≤ 0. 

Answer 1

The congruence relation may be rewritten as

$$a=23k+8, where \space k-integer$$

if

$$-22\leq a \leq 0$$

then

$$-22\leq 23k+8 \leq 0$$ $$-30\leq 23k \leq -8$$

so

$$k=-1$$

and

$$a=23k+8=23(-1)+8=-15$$ $$-15\equiv8(mod 23)\space and \space -22\leq -15 \leq 0$$

Answer: a=-15