Hypothesis, conclusion and counterexample

Statement

Consider the statement: “If $a\mid b$ and $c\mid d$, then $(a+c)\mid(b+d)$”. Identify the hypothesis and the conclusion, then decide whether the statement is true by giving an example or a counterexample.

Solution

Hypothesis: ”$a\mid b$ and $c\mid d$”. Conclusion: "$(a+c)\mid(b+d)$".

The statement is false. A counterexample: $a=2,\ b=8,\ c=3,\ d=9$. Then $2\mid 8$ and $3\mid 9$, but $a+c = 5, \qquad b+d = 17, \qquad 5 \nmid 17.$ A single counterexample is enough to make the implication false.