Question

which is a way to use prime factorization to find the least common multiple of 8 and 6?
6 → 2×3
8 → 2×2×2
lcm = 2×2×2×3
=24

6 → 2×3
8 → 2×2×2
lcm = 2

6 → 2×3
8 → 2×2×2
lcm = 2×2×2×2×3
=48

6 → 2×3
8 → 2×2×2
lcm = 2×2×3
=12

Explanation:

Step1: Recall LCM via Prime Factorization

To find LCM using prime factorization, we take the highest power of each prime number from the factorizations of the given numbers and multiply them. For 6 and 8:

• Prime factorization of 6: \( 6 = 2 \times 3 \)
• Prime factorization of 8: \( 8 = 2 \times 2 \times 2 = 2^3 \)

Step2: Analyze Each Option

• Top - Left Option:
• Prime factors: 6 is \( 2 \times 3 \), 8 is \( 2 \times 2 \times 2 \).
• LCM is calculated as \( 2^3 \times 3 = 2 \times 2 \times 2 \times 3 = 24 \), which follows the rule (highest power of 2 is \( 2^3 \), highest power of 3 is \( 3^1 \)).
• Top - Right Option: LCM is given as 2, which is incorrect (2 is a common factor, not LCM).
• Bottom - Left Option: LCM is \( 2^4 \times 3 = 48 \), but we only need \( 2^3 \) (from 8), so this is incorrect.
• Bottom - Right Option: LCM is \( 2^2 \times 3 = 12 \), but we need \( 2^3 \) (from 8), so this is incorrect.

Answer:

The option with "6 ➔ 2 × 3, 8 ➔ 2 × 2 × 2, LCM = 2 × 2 × 2 × 3 = 24" (the top - left option)