What is the least common multiple of 14 and 15?

Steps to Solve

1. Identify the prime factorization of each number

To find the least common multiple (LCM), start by finding the prime factorization of both numbers.

For 14:
$14 = 2^1 \times 7^1$

For 15:
$15 = 3^1 \times 5^1$

2. List all prime factors with the highest powers

Next, we create a list of all unique prime factors from the factorizations and select the highest power of each factor.

• From 14: $2^1$ and $7^1$
• From 15: $3^1$ and $5^1$

The unique prime factors are $2$, $3$, $5$, and $7$.

3. Combine the prime factors

The LCM is obtained by multiplying each prime factor raised to their highest power:

$$ \text{LCM} = 2^1 \times 3^1 \times 5^1 \times 7^1 $$

4. Calculate the LCM

Now calculate the product:

[ \text{LCM} = 2 \times 3 \times 5 \times 7 ]

Calculating this step-by-step:
First, multiply $2$ and $3$:
$2 \times 3 = 6$

Next, multiply $6$ by $5$:
$6 \times 5 = 30$

Finally, multiply $30$ by $7$:
$30 \times 7 = 210$

5. Final Result

The least common multiple of 14 and 15 is:

$$ \text{LCM} = 210 $$