Factors, Multiples, Primes and Composites

$1, 2, 5, 10$

$1 \times 10$ and $2 \times 5$.

$1, 2, 4, 8, 16$

$1 \times 16$, $2 \times 8$ and $4 \times 4$. The repeated pair gives an odd count.

Yes

$30 \div 6 = 5$ exactly.

$8, 16, 24, 32$

Multiply $8$ by $1$, $2$, $3$ and $4$.

Composite

Its factors are $1$, $3$ and $9$ — three of them, which is more than two.

Yes

Nothing between $1$ and $13$ divides it.

Yes

Its only factors are $1$ and $2$. It is the only even prime, since every other even number also has $2$ as a factor.

Work up from $1$, recording both partners.

$1, 2, 3, 4, 6, 8, 12, 24$

$1 \times 24$, $2 \times 12$, $3 \times 8$, $4 \times 6$.

Testing $5$ fails, and by $6$ the pairs have met, so the list is complete.

Because $6 \times 6 = 36$, so one pair is a number with itself.

Factors normally come in distinct pairs, giving an even count.

A perfect square has one pair where both members are the same number, so it is counted once instead of twice. That is what makes the total odd, and perfect squares are the only numbers it happens to.

How many factors does a prime need?

No. A prime needs exactly two factors, and $1$ has one.

His observation is correct — nothing divides $1$ except itself. The definition asks for something slightly different.

A prime has exactly two factors, itself and $1$. For the number $1$, those are the same number, so it has only one factor.

The exclusion is not arbitrary. Every whole number breaks into primes in exactly one way — $12$ is $2 \times 2 \times 3$ and nothing else. If $1$ counted as prime, $12$ would also be $1 \times 2 \times 2 \times 3$, and $1 \times 1 \times 2 \times 2 \times 3$, and so on forever. Leaving $1$ out is what keeps that uniqueness.