23. Exemplar exercise problems VIII

1. A positive integer is of the form $3q + 1$, $q$ being a natural number. Can you write its square in any form other than $3m + 1$, i.e., $3m$ or $3m + 2$ for some integer $m$? Justify your answer.

 

2. Can two numbers have $18$ as their HCF and $380$ as their LCM? Give reasons.

 

3. Explain why $3 \times 5 \times 7 + 7$ is a composite number.