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.