lcm of 14 and 42

Steps to Solve

1. Determine the Prime Factorizations First, we need to find the prime factorization of both numbers.

For 14:

• 14 can be divided by 2 (which is prime): $14 = 2 \times 7$.

For 42:

• 42 can be divided by 2 (which is prime): $42 = 2 \times 21$.
• 21 can be further divided by 3 (which is prime): $21 = 3 \times 7$. Thus, $42 = 2 \times 3 \times 7$.

2. Identify the Unique Prime Factors List the unique prime factors from both factorizations:

• From 14: 2, 7
• From 42: 2, 3, 7

The unique prime factors are: 2, 3, and 7.

3. Use Each Prime Factor the Maximum Number of Times Found For the LCM, take the highest power of each prime factor:

• For 2, the maximum power is $2^1$ (from both 14 and 42).
• For 3, the maximum power is $3^1$ (from 42).
• For 7, the maximum power is $7^1$ (from both).

4. Calculate the LCM Now, multiply all these together: $$ LCM = 2^1 \times 3^1 \times 7^1 = 2 \times 3 \times 7 $$ Calculating this: $$ 2 \times 3 = 6 \ 6 \times 7 = 42 $$

The least common multiple (LCM) of 14 and 42 is 42.