Question

(a) If $k\equiv 1\left(mod4\right)$, then what is $6k+5$congruent to modulo 4 ?

(b) If $r\equiv 3\left(mod10\right)$and $s\equiv 7\left(mod10\right)$, then what is $2r+3s$ congruent to modulo 10?

Short answer

1. It is obtained that $6k+5$ is congruent to 3 modulo 4.
2. It is obtained that $2r+3s$is congruent to 5 modulo 10.

Step-by-step solution

Write the definition of Congruence

Consider a, b, n as an integer with $n>0$ . If ndivides , then a is congruent to b modulo n (expressed as $a\equiv b\left(modn\right)$).

Determine 6k+5 congruent to modulo 4 

a)

When $k\equiv 1\left(mod4\right)$then $6k\equiv 6·1\equiv 2\left(mod4\right)$.

The value of$6k+5$ can be expressed as follows:

$\begin{array}{rcl}6k+5& =& 2+5\\ & =& 3\left(mod4\right)\end{array}$

Thus, it is obtained that $6k+5$ iscongruentto 3 modulo 4.

Determine 2r+3s congruent to modulo 

b)

When $r\equiv 3\left(mod10\right)$and $s\equiv 7\left(mod10\right)$, then there is an integer p, q as follows:

$\begin{array}{rcl}r-3& =& 10p\\ r& =& 10p+3\end{array}$

$\begin{array}{rcl}s+7& =& 10q\\ s& =& 10q-7\end{array}$

Substitute the value of r and s into the expression $2r+3s$ as follows:

role="math" localid="1646285767835" $\begin{array}{rcl}2r+3s& =& 2\left(10p+3\right)+3\left(10q-7\right)\\ & =& 20p+6+30q-21\\ & =& 10\left(2p+3q\right)-15\\ & =& 10\left(2p+3q-2\right)-5\end{array}$

Since $2p+3q-2$ is an integer, $2r+3s\equiv 5\left(10\right)$.

Thus, it is obtained that$2r+3s$ iscongruentto 5 modulo 10.