The Distributive Property: Practice

The greatest common factor of $12$ and $18$ is $6$, since $12 = 6 \times 2$ and $18 = 6 \times 3$.

$12 + 18 = 6 \times 2 + 6 \times 3 = 6(2 + 3)$

Check by distributing back: $6(2 + 3) = 12 + 18 = 30$. The other options give $36$, $45$, and $120$.

The greatest common factor of $20$ and $35$ is $5$, since $20 = 5 \times 4$ and $35 = 5 \times 7$.

$20 + 35 = 5 \times 4 + 5 \times 7 = 5(4 + 7)$

Check: $5(4 + 7) = 5 \times 11 = 55$, and $20 + 35 = 55$.

Distribute and add each option, then find the one that misses $24$.

$2(7 + 4) = 2 \times 11 = 22 \ne 24$

The others give $4 \times 6 = 24$, $3 \times 8 = 24$, and $6 \times 4 = 24$.

Write $102 = 100 + 2$ and distribute the $12$.

$12 \times 102 = 12(100 + 2) = 12 \times 100 + 12 \times 2 = 1200 + 24 = 1224$

The greatest common factor of $30$ and $45$ is $15$, since $30 = 15 \times 2$ and $45 = 15 \times 3$.

$30 + 45 = 15 \times 2 + 15 \times 3 = 15(2 + 3)$

Check: $15(2 + 3) = 15 \times 5 = 75$, and $30 + 45 = 75$. The other options give $165$, $90$, and $705$.

Distribution spreads over addition and subtraction, not over a product. With a product inside, just multiply.

$4(3 \times 5) = 4 \times 15 = 60$

Distributing it as $4 \times 3 + 4 \times 5 = 32$ would be wrong; the parentheses hold a product, not a sum.

Split $21$ as $20 + 1$ and distribute the $15$.

$15 \times 21 = 15(20 + 1) = 15 \times 20 + 15 \times 1 = 300 + 15 = 315$

The greatest common factor of $16$ and $40$ is $8$, since $16 = 8 \times 2$ and $40 = 8 \times 5$.

$16 + 40 = 8 \times 2 + 8 \times 5 = 8(2 + 5)$

Check: $8(2 + 5) = 8 \times 7 = 56$, and $16 + 40 = 56$. The other options give $64$, $80$, and $336$.

Write $104 = 100 + 4$ and distribute the $25$.

$25 \times 104 = 25(100 + 4) = 25 \times 100 + 25 \times 4 = 2500 + 100 = 2600$

Multiply each of the three terms by $4$, then add.

$4(10 + 5 + 2) = 4 \times 10 + 4 \times 5 + 4 \times 2 = 40 + 20 + 8 = 68$

Check: $4 \times 17 = 68$.

Write $296 = 300 - 4$ and distribute the $7$ to each part.

$7 \times 296 = 7(300 - 4) = 7 \times 300 - 7 \times 4 = 2100 - 28 = 2072$

Subtracting only $4$ leaves $2096$, which is wrong.

Test each option by distributing back. Only $12(2 + 3)$ and $6(4 + 6)$ return $24 + 36 = 60$; the other two give $456$ and $54$, so they are not even true.

$24 + 36 = 6 \times 4 + 6 \times 6 = 6(4 + 6)$

That equation is true, but the factoring is unfinished: $4$ and $6$ still share a factor of $2$. Pulling that out too gives the complete form $12(2 + 3)$.