Prove that if 6 divides a², then 6 divides a…

Step 1. Part (a), the primes of 6. If $6$ divides $a^2$, then the primes $2$ and $3$ both divide $a^2$.

Step 2. Use the key fact twice. $2$ divides $a$, and $3$ divides $a$. Since $2$ and $3$ are different primes, their product $6$ divides $a$.

Step 3. Part (b), test small values. $a = 1, 2, 3$ give $a^2 = 1, 4, 9$, and $8$ divides none of them.

Step 4. The first failure. $a = 4$ gives $a^2 = 16$, and $8$ divides $16$. But $8$ does not divide $4$.