Prime Factorization: Practice

Divide by the smallest prime each time: $18 \div 2 = 9$, then $9 = 3 \times 3$.

$18 = 2 \times 3 \times 3 = 2 \times 3^2$

The form $2^2 \times 3 = 12$ has the wrong exponents, and $3 \times 6$ leaves $6$ composite.

Peel off $2$s: $48 \div 2 = 24$, $24 \div 2 = 12$, $12 \div 2 = 6$, $6 \div 2 = 3$, and $3$ is prime.

$48 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3$

Four $2$s, not three, so $2^3 \times 3 = 24$ is too small.

It is odd with digit sum $9$, so divide by $3$: $63 \div 3 = 21$, then $21 \div 3 = 7$.

$63 = 3 \times 3 \times 7 = 3^2 \times 7$

The form $3 \times 7^2 = 147$ swaps which prime is squared.

Keep dividing by $2$: $96 \to 48 \to 24 \to 12 \to 6 \to 3$, which is five halvings, then a $3$.

$96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 = 2^5 \times 3$

So there are five $2$s; $2^4 \times 3 = 48$ is too small and $2^5 = 32$ drops the $3$.

It is odd with digit sum $12$, so divide by $3$: $75 \div 3 = 25$, and $25 = 5 \times 5$.

$75 = 3 \times 5 \times 5 = 3 \times 5^2$

The form $3^2 \times 5 = 45$ squares the wrong prime.

Divide by $2$, then factor what remains: $98 \div 2 = 49$, and $49 = 7 \times 7$.

$98 = 2 \times 7 \times 7 = 2 \times 7^2$

The forms $2 \times 49$ and $7 \times 14$ keep a composite factor.

Group the equal leaves and count each prime: two $2$s and two $3$s.

$36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$

The form $4 \times 9$ is the right value but not built from primes, and the other two miscount the exponents.

Divide by $2$ once, then by $3$ repeatedly: $54 \div 2 = 27$, $27 \div 3 = 9$, $9 \div 3 = 3$, $3 \div 3 = 1$.

$54 = 2 \times 3 \times 3 \times 3 = 2 \times 3^3$

The form $2 \times 27$ leaves $27$ composite, and $2^2 \times 3^3 = 108$.

Halve down to an odd number, then take the $3$s: $144 \to 72 \to 36 \to 18 \to 9$, and $9 = 3 \times 3$.

$144 = 2^4 \times 3^2$

That is four $2$s and two $3$s; $12^2$ and $2^4 \times 9$ are not written with primes.

Divide by the small primes: $66 \div 2 = 33$, $33 \div 3 = 11$, and $11$ is prime.

$66 = 2 \times 3 \times 11$

Each prime appears once, so there are no exponents above $1$; $2^2 \times 3 \times 11 = 132$ is twice too big.

Divide: $126 \div 2 = 63$, $63 \div 3 = 21$, $21 \div 3 = 7$, $7 \div 7 = 1$.

$126 = 2 \times 3 \times 3 \times 7 = 2 \times 3^2 \times 7$

The form $2^2 \times 3^2 \times 7 = 252$ has an extra factor of $2$.

Divide: $90 \div 2 = 45$, $45 \div 3 = 15$, $15 \div 3 = 5$, $5 \div 5 = 1$.

$90 = 2 \times 3 \times 3 \times 5 = 2 \times 3^2 \times 5$

The form $2^2 \times 3 \times 5 = 60$ has one $2$ too many and is missing a $3$.