Chapter 3, Problem 1SE

To determine

If $37\equiv 18\left(\mathrm{mod}2\right)$ is true or false.

Answer to Problem 1SE

The congruence $37\equiv 18\left(\mathrm{mod}2\right)$ is false.

Explanation of Solution

Procedure used:

Procedure to check the divisibility.

1 Consider the syntax congruence provided.

2 Find if the congruence is viable.

3 Obtain and dictate the answer

Viability of congruence.

The congruence defined as $x=y\left(\mathrm{mod}m\right)$ is true if $x-y$ is divisible by m.

Calculation:

Consider the given congruence as below.

$37\equiv 18\left(\mathrm{mod}2\right)$

The congruence defined as $x=y\left(\mathrm{mod}m\right)$ is true if $x-y$ is divisible by m. Therefore, check if $37-18$ is divisible by 2. Thus,

$\begin{array}{l}37-18=19;\\ 19=9\cdot 2+1\end{array}$

Since $r=1$, the remainder is nonzero. Therefore, $2$ does not divide $19$. Hence, $37\cancel{\equiv }18\left(\mathrm{mod}2\right)$.

Therefore, the congruence $37\equiv 18\left(\mathrm{mod}2\right)$ is false.