GCF and LCM: Practice

Write $84 = 2^2 \times 3 \times 7$ and $126 = 2 \times 3^2 \times 7$. Shared primes are $2, 3, 7$; take the lower power of each: $2^1$, $3^1$, $7^1$.

$\operatorname{GCF}(84, 126) = 2 \times 3 \times 7 = 42$

Each of $2$, $3$, and $7$ is taken to the smaller of its two powers.

By the product rule for two numbers, $a \times b = \operatorname{GCF}(a, b) \times \operatorname{LCM}(a, b) = 8 \times 96 = 768$. Now divide by $a = 24$.

$b = 768 \div 24 = 32$

Check: $\operatorname{GCF}(24, 32) = 8$ and $\operatorname{LCM}(24, 32) = 96$, as required.

Write $90 = 2 \times 3^2 \times 5$ and $135 = 3^3 \times 5$. Shared primes are $3$ and $5$; take the lower power of each: $3^2$ and $5^1$.

$\operatorname{GCF}(90, 135) = 3^2 \times 5 = 9 \times 5 = 45$

The prime $2$ is only in $90$, so it is not part of the GCF.

Write $14 = 2 \times 7$ and $21 = 3 \times 7$. Take the highest power of each prime: $2, 3, 7$.

$\operatorname{LCM}(14, 21) = 2 \times 3 \times 7 = 42$

Because they share the factor $7$, the LCM ($42$) is smaller than the plain product ($294$).

Write $100 = 2^2 \times 5^2$ and $250 = 2 \times 5^3$. Shared primes are $2$ and $5$; take the lower power of each: $2^1$ and $5^2$.

$\operatorname{GCF}(100, 250) = 2 \times 5^2 = 2 \times 25 = 50$

The prime $5$ appears twice in $100$ and three times in $250$, so the GCF keeps two copies.

Write $16 = 2^4$ and $18 = 2 \times 3^2$. Take the highest power of each prime: $2^4$ and $3^2$.

$\operatorname{LCM}(16, 18) = 2^4 \times 3^2 = 16 \times 9 = 144$

The value $2$ is the GCF, and $288 = 16 \times 18$ is the plain product.

Both numbers must be multiples of $12$, and their GCF must be exactly $12$ (not larger). Check $24$ and $36$: both are multiples of $12$, and $\operatorname{GCF}(24, 36) = 12$.

$24 = 12 \times 2, \qquad 36 = 12 \times 3, \qquad \operatorname{GCF}(2, 3) = 1$

Since $24$ and $36$ are $12$ times two numbers with no common factor, their GCF is exactly $12$. The other pairs include a number that is not a multiple of $12$.

Write $66 = 2 \times 3 \times 11$ and $99 = 3^2 \times 11$. Shared primes are $3$ and $11$; take the lower power of each: $3^1$ and $11^1$.

$\operatorname{GCF}(66, 99) = 3 \times 11 = 33$

The prime $2$ is only in $66$, so it is left out of the common factor.

Write $9 = 3^2$, $12 = 2^2 \times 3$, and $15 = 3 \times 5$. Take the highest power of each prime across all three: $2^2$, $3^2$, $5$.

$\operatorname{LCM}(9, 12, 15) = 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180$

Every prime is taken to its largest appearance, so all three numbers divide $180$.

Work directly from the prime powers, taking the lower power of each shared prime: $2^2$ (the lower of $2^3$ and $2^2$), $3$ (lower of $3^2$ and $3^1$), $5$ (lower of $5^1$ and $5^2$).

$\operatorname{GCF} = 2^2 \times 3 \times 5 = 60$

Reading the answer off the prime powers is faster than multiplying the numbers out first.

Take the higher power of each prime: $2^3$ (the higher of $2^3$ and $2^2$), $3^2$ (higher of $3^2$ and $3^1$), $5^2$ (higher of $5^1$ and $5^2$).

$\operatorname{LCM} = 2^3 \times 3^2 \times 5^2 = 8 \times 9 \times 25 = 1800$

You pick the larger power for each prime; you never add the exponents together.

They leave together at a common multiple of $15$ and $25$, and next at the least one. Write $15 = 3 \times 5$ and $25 = 5^2$, then take the highest power of each prime.

$\operatorname{LCM}(15, 25) = 3 \times 5^2 = 3 \times 25 = 75$

So they next leave together after $75$ minutes, which is $1$ hour $15$ minutes after $8{:}00$, at $9{:}15$.