Question

What is the least common multiple of 70, 60, and 50 ?
F.
G.
Н.
60
180
210
2,100
K. 210,000

Answer 1

To find the least common multiple (LCM) of the numbers 70, 60, and 50, we first determine their prime factorizations:

- **70**: \( 2 \times 5 \times 7 \)
- **60**: \( 2^2 \times 3 \times 5 \)
- **50**: \( 2 \times 5^2 \)

Next, we take the highest power of each prime that appears in the factorizations:

- For \( 2 \): The highest power is \( 2^2 \) (from 60).
- For \( 3 \): The highest power is \( 3^1 \) (from 60).
- For \( 5 \): The highest power is \( 5^2 \) (from 50).
- For \( 7 \): The highest power is \( 7^1 \) (from 70).

Now, we multiply these highest powers together to get the LCM:

\[
LCM = 2^2 \times 3^1 \times 5^2 \times 7^1
\]

Calculating each step:

1. \( 2^2 = 4 \)
2. \( 3^1 = 3 \)
3. \( 5^2 = 25 \)
4. \( 7^1 = 7 \)

Now we multiply these results together:

\[
LCM = 4 \times 3 \times 25 \times 7
\]

Calculating \( 4 \times 3 = 12 \):

\[
12 \times 25 = 300
\]

Then calculating \( 300 \times 7 = 2100 \):

\[
LCM = 2100
\]

Thus, the least common multiple of 70, 60, and 50 is \( \boxed{2100} \).