Question

extra practice – factor trees
objective: students will be able to use modeling and visual skills to find the prime factorization of a number.
fill in the missing numbers for each of the factor trees.
18
/\\
3 × 6
/\\
□ × 3
factors:
18: 2 × 3 × 3
= 2 × 3²
64
/\\
8 × 8
/\\ /\\
□ × 4 2×□
/\\ /\\
□×□ 2×□
factors:
64: 2 × □ × 2 × □ × □ × □ = 2^□
88
/\\
4×□
/\\ /\\
□×2 2×□
factors:
88: 2 × 2 × 2 × □
= 2³ × 11
fill in the missing numbers for each of the factor trees.
60
/\\
6 × □
/\\ /\\
□×3 2×□
60
/\\
□×□
/\\
5×□
/\\
2 × 3
factors: 60: 2 × □ × 3 × □ = □² × 3 × □

Explanation:

Step1: Factor 6 into primes

$6 = 2 \times 3$

Step2: Factor 8 into primes (first 64 tree)

$8 = 2 \times 4$

Step3: Factor 4 into primes

$4 = 2 \times 2$

Step4: Factor second 8 in 64 tree

$8 = 2 \times 4$, $4 = 2 \times 2$

Step5: Count 2s for 64's exponent

There are 6 factors of 2, so $2^6$

Step6: Factor 88's top pair

$88 = 4 \times 22$

Step7: Factor 4 into primes

$4 = 2 \times 2$

Step8: Factor 22 into primes

$22 = 2 \times 11$

Step9: Factor 60's first top pair

$60 = 6 \times 10$

Step10: Factor 6 into primes

$6 = 2 \times 3$

Step11: Factor 10 into primes

$10 = 2 \times 5$

Step12: Factor 60's second top pair

$60 = 10 \times 6$

Step13: Factor 6 into primes

$6 = 2 \times 3$

Step14: Write 60's prime factorization

$60 = 2^2 \times 3 \times 5$

Answer:

For 18:

• Empty box under 6: $\boldsymbol{2}$

For 64:

• Empty box under first 8: $\boldsymbol{2}$
• Empty boxes under first 4: $\boldsymbol{2}$, $\boldsymbol{2}$
• Empty box under second 8: $\boldsymbol{4}$
• Empty box under second 4: $\boldsymbol{2}$
• Empty boxes in factors line: $\boldsymbol{2}$, $\boldsymbol{2}$, $\boldsymbol{2}$, $\boldsymbol{2}$
• Exponent box: $\boldsymbol{6}$

For 88:

• Empty box next to 4: $\boldsymbol{22}$
• Empty box under 4: $\boldsymbol{2}$
• Empty box under 22: $\boldsymbol{11}$
• Empty box in factors line: $\boldsymbol{11}$

For first 60 tree:

• Empty box next to 6: $\boldsymbol{10}$
• Empty box under 6: $\boldsymbol{2}$
• Empty box under 10: $\boldsymbol{5}$

For second 60 tree:

• Empty boxes at the top: $\boldsymbol{10}$, $\boldsymbol{6}$
• Empty box under 6: $\boldsymbol{6}$

For 60's factors line:

• Empty boxes (left to right): $\boldsymbol{2}$, $\boldsymbol{5}$, $\boldsymbol{2}$, $\boldsymbol{5}$