Question

which choice shows how to find the least common multiple of 9 and 42 through prime factorization?
9 → 3 × 3
42 → 2 × 3 × 7
lcm: 2 × 3 × 3 × 7 = 126
9 → 3 × 3
42 → 2 × 3 × 7
lcm: 3 × 3 × 2 × 3 × 7 = 378
9 → 3 × 3
42 → 2 × 3 × 7
lcm: 3
9 → 3 × 3
42 → 2 × 3 × 7
lcm: 2 × 3 × 7 = 42

Explanation:

Step1: Prime Factorize 9 and 42

Prime factorization of \( 9 = 3\times3 \) and \( 42 = 2\times3\times7 \).

Step2: Find LCM via Prime Factors

To find LCM, take the highest power of each prime factor present. For prime 2: \( 2^1 \), for prime 3: \( 3^2 \) (from 9), for prime 7: \( 7^1 \). Multiply them: \( 2\times3\times3\times7 = 126 \).
Check the options: The first option (top - left) has \( 9 = 3\times3 \), \( 42 = 2\times3\times7 \), and LCM calculated as \( 2\times3\times3\times7 = 126 \), which matches the correct method. The other options either repeat factors (second option), take GCD (third option), or miss the highest power of 3 (fourth option).

Answer:

The option with \( 9 \to 3\times3 \), \( 42 \to 2\times3\times7 \), and \( \text{LCM: } 2\times3\times3\times7 = 126 \) (the top - left option among the four choices).