Question

solve for n

1. $3 \times (2 + n) = (3 \times 2) + (3 \times n)$ $n = \underline{\quad\quad\quad\quad\quad\quad\quad\quad}$
2. $2 \times (n + 5) = 12 + 10$ $n = \underline{\quad\quad\quad\quad\quad\quad\quad\quad}$

find the greatest common factor of each set of numbers.

1. 4 and 6 $\underline{\quad\quad\quad\quad\quad\quad\quad\quad}$ 8. 24 and 15 $\underline{\quad\quad\quad\quad\quad\quad\quad\quad}$
2. 27 and 18 $\underline{\quad\quad\quad\quad\quad\quad\quad\quad}$ 10. 14 and 42 $\underline{\quad\quad\quad\quad\quad\quad\quad\quad}$

find the least common multiple of each set of numbers.

1. 42 and 63 $\underline{\quad\quad\quad\quad\quad\quad\quad\quad}$ 12. 3 and 8 $\underline{\quad\quad\quad\quad\quad\quad\quad\quad}$
2. 12 and 15 $\underline{\quad\quad\quad\quad\quad\quad\quad\quad}$ 14. 9, 10, and 45 $\underline{\quad\quad\quad\quad\quad\quad\quad\quad}$

Explanation:

Step1: Identify distributive property

$3 \times (2 + n) = (3 \times 2) + (3 \times n)$ matches $a(b+c)=ab+ac$, so $n$ is the missing term.

Step2: Simplify right-hand side (RHS)

$12 + 10 = 22$

Step3: Isolate the parenthetical term

Divide both sides by 2: $\frac{2 \times (n+5)}{2} = \frac{22}{2}$ → $n+5=11$

Step4: Solve for $n$

Subtract 5 from both sides: $n=11-5=6$

Step5: List factors of 4 and 6

Factors of 4: $1,2,4$; Factors of 6: $1,2,3,6$. The largest shared factor is 2.

Step6: List factors of 24 and 15

Factors of 24: $1,2,3,4,6,8,12,24$; Factors of 15: $1,3,5,15$. The largest shared factor is 3.

Step7: List factors of 27 and 18

Factors of 27: $1,3,9,27$; Factors of 18: $1,2,3,6,9,18$. The largest shared factor is 9.

Step8: List factors of 14 and 42

Factors of 14: $1,2,7,14$; Factors of 42: $1,2,3,6,7,14,21,42$. The largest shared factor is 14.

Step9: Prime factorize 42 and 63

$42=2 \times 3 \times 7$, $63=3^2 \times 7$. LCM is $2 \times 3^2 \times 7=126$

Step10: Use coprime LCM rule

3 and 8 are coprime, so LCM = $3 \times 8=24$

Step11: Prime factorize 12 and 16

$12=2^2 \times 3$, $16=2^4$. LCM is $2^4 \times 3=48$

Step12: Prime factorize 9,10,45

$9=3^2$, $10=2 \times 5$, $45=3^2 \times 5$. LCM is $2 \times 3^2 \times 5=90$

Answer:

1. $n = 6$
2. $n = 6$
3. 2
4. 3
5. 9
6. 14
7. 126
8. 24
9. 48
10. 90