GCF and LCM: Practice

For the GCF, take the lowest power of each shared prime. For $2$: lowest of $2^1$ and $2^3$ is $2^1$. For $3$: lowest of $3^2$ and $3^1$ is $3^1$.

$\operatorname{GCF}(18, 24) = 2 \times 3 = 6$

The value $72$ is the LCM, taken from the highest powers instead.

For the LCM, take the highest power of each prime. For $2$: highest of $2^1$ and $2^3$ is $2^3$. For $3$: highest of $3^2$ and $3^1$ is $3^2$.

$\operatorname{LCM}(18, 24) = 2^3 \times 3^2 = 8 \times 9 = 72$

The value $6$ is the GCF, taken from the lowest powers.

Write $16 = 2^4$ and $24 = 2^3 \times 3$. The only shared prime is $2$; take the lower power, $2^3$.

$\operatorname{GCF}(16, 24) = 2^3 = 8$

The prime $3$ is only in $24$, so it does not appear in the GCF.

Write $6 = 2 \times 3$ and $10 = 2 \times 5$. Take the highest power of each prime: $2, 3,$ and $5$.

$\operatorname{LCM}(6, 10) = 2 \times 3 \times 5 = 30$

The plain product $6 \times 10 = 60$ is a common multiple, but not the least one, because $6$ and $10$ share the factor $2$.

Write $8 = 2^3$ and $12 = 2^2 \times 3$. Take the highest power of each prime: $2^3$ and $3$.

$\operatorname{LCM}(8, 12) = 2^3 \times 3 = 8 \times 3 = 24$

The value $4$ is the GCF, and $96 = 8 \times 12$ is the plain product.

Write $14 = 2 \times 7$ and $35 = 5 \times 7$. The only shared prime is $7$.

$\operatorname{GCF}(14, 35) = 7$

The primes $2$ and $5$ each appear in only one number, so neither is part of the common factor.

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

$\operatorname{LCM}(9, 15) = 3^2 \times 5 = 9 \times 5 = 45$

The value $3$ is the GCF, and $135 = 9 \times 15$ is the plain product.

The product of two numbers equals the product of their GCF and LCM.

$a \times b = \operatorname{GCF}(a, b) \times \operatorname{LCM}(a, b) = 4 \times 60 = 240$

So whatever $a$ and $b$ are, their product must be $240$.

They blink together at a common multiple of $4$ and $6$, and next at the least one. Write $4 = 2^2$ and $6 = 2 \times 3$, then take the highest power of each prime.

$\operatorname{LCM}(4, 6) = 2^2 \times 3 = 4 \times 3 = 12$

So they next blink together after $12$ seconds. The value $2$ is the GCF, not the answer.

Write $21 = 3 \times 7$ and $28 = 2^2 \times 7$. The only shared prime is $7$.

$\operatorname{GCF}(21, 28) = 7$

The primes $3$ and $2$ each appear in only one number, so they are not part of the common factor.

Write $12 = 2^2 \times 3$ and $16 = 2^4$. Take the highest power of each prime: $2^4$ and $3$.

$\operatorname{LCM}(12, 16) = 2^4 \times 3 = 16 \times 3 = 48$

The value $4$ is the GCF, and $192 = 12 \times 16$ is the plain product.

The product rule, which is about two numbers, says $\operatorname{LCM}(a, b) = (a \times b) \div \operatorname{GCF}(a, b)$. Substitute $a = 18$, $b = 24$, and $\operatorname{GCF}(a, b) = 6$.

$\operatorname{LCM}(18, 24) = (18 \times 24) \div 6 = 432 \div 6 = 72$

The value $432$ is the plain product, before dividing by the GCF.