Which pairs of numbers have an LCM of 24? Choose ALL that apply.
A. 3 and 8 B. 2 and 12 C. 6 and 4 D. 8 and 6 E. 4 and 12
The Correct Answer and Explanation is :
To find which pairs of numbers have a least common multiple (LCM) of 24, we can calculate the LCM for each given pair.
A. 3 and 8
- The prime factors are:
- 3 = (3^1)
- 8 = (2^3)
- LCM is calculated by taking the highest power of each prime factor:
[
\text{LCM}(3, 8) = 3^1 \times 2^3 = 3 \times 8 = 24
] - Result: LCM is 24.
B. 2 and 12
- The prime factors are:
- 2 = (2^1)
- 12 = (2^2 \times 3^1)
- LCM is:
[
\text{LCM}(2, 12) = 2^2 \times 3^1 = 4 \times 3 = 12
] - Result: LCM is 12 (not 24).
C. 6 and 4
- The prime factors are:
- 6 = (2^1 \times 3^1)
- 4 = (2^2)
- LCM is:
[
\text{LCM}(6, 4) = 2^2 \times 3^1 = 4 \times 3 = 12
] - Result: LCM is 12 (not 24).
D. 8 and 6
- The prime factors are:
- 8 = (2^3)
- 6 = (2^1 \times 3^1)
- LCM is:
[
\text{LCM}(8, 6) = 2^3 \times 3^1 = 8 \times 3 = 24
] - Result: LCM is 24.
E. 4 and 12
- The prime factors are:
- 4 = (2^2)
- 12 = (2^2 \times 3^1)
- LCM is:
[
\text{LCM}(4, 12) = 2^2 \times 3^1 = 4 \times 3 = 12
] - Result: LCM is 12 (not 24).
Summary of Results:
- Pairs with LCM of 24: A. 3 and 8 and D. 8 and 6.
- Pairs with LCM not equal to 24: B. 2 and 12, C. 6 and 4, E. 4 and 12.
The key to determining the LCM involves using the highest powers of all prime factors in each pair. The pairs that meet the criteria of having an LCM of 24 are those that produce the required multiplicative combination of prime factors. Thus, the correct answers are A and D.