Question

Find four numbers congruent 5modulo 17.

Short answer

The four possible numbers are 22, 39, 56, 73.

Step-by-step solution

Step 1

5 mod 17

All congruent numbers modulo 17 differ by a multiple of 17 , thus all numbers congruent to 5 modulo 17 are 5 increased by multiple of 17.

$5mod17\equiv 5+17kmod17(\text{for every integer}k\text{)}$

Let us evaluate 5 + 17k for a few values of k , for example k = 1,2,3,4

$\begin{array}{l}k=1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}5+17k=5+17\left(1\right)=22\\ k=2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}5+17k=5+17\left(2\right)=39\\ k=3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}5+17k=5+17\left(3\right)=56\\ k=4\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}5+17k=5+17\left(4\right)=73\end{array}$

Thus four possible numbers are then 22, 39, 56, 73.

Note: If you use different k - values, then you obtain four different numbers.