To assess whether a student can correctly find the least common multiple (LCM) of three numbers, it's crucial to use a set of numbers where the calculation process involves multiple steps and different prime factors.
Given the provided sets of numbers, let's determine which set would be the best for this assessment:
1. Set 1: [tex]\( 14, 30, 44 \)[/tex]
- Prime factorization:
- [tex]\( 14 = 2 \times 7 \)[/tex]
- [tex]\( 30 = 2 \times 3 \times 5 \)[/tex]
- [tex]\( 44 = 2^2 \times 11 \)[/tex]
2. Set 2: [tex]\( 22, 33, 55 \)[/tex]
- Prime factorization:
- [tex]\( 22 = 2 \times 11 \)[/tex]
- [tex]\( 33 = 3 \times 11 \)[/tex]
- [tex]\( 55 = 5 \times 11 \)[/tex]
3. Set 3: [tex]\( 36, 54, 63 \)[/tex]
- Prime factorization:
- [tex]\( 36 = 2^2 \times 3^2 \)[/tex]
- [tex]\( 54 = 2 \times 3^3 \)[/tex]
- [tex]\( 63 = 3^2 \times 7 \)[/tex]
4. Set 4: [tex]\( 28, 70, 126 \)[/tex]
- Prime factorization:
- [tex]\( 28 = 2^2 \times 7 \)[/tex]
- [tex]\( 70 = 2 \times 5 \times 7 \)[/tex]
- [tex]\( 126 = 2 \times 3^2 \times 7 \)[/tex]
To properly assess the student's understanding of finding the LCM, it's beneficial to use a set with varied factors and more complex calculations. Among the provided sets, set 3 ([tex]\( 36, 54, 63 \)[/tex]) stands out as the best choice because:
1. It includes multiple prime factors: 2, 3, and 7.
2. It involves different powers of the primes, which students need to correctly identify and use in their LCM calculation.
Let's verify the LCM of set 3 ([tex]\( 36, 54, 63 \)[/tex]) step-by-step.
### Step-by-Step Calculation of LCM for Set 3:
1. Prime Factorization:
- [tex]\( 36 = 2^2 \times 3^2 \)[/tex]
- [tex]\( 54 = 2 \times 3^3 \)[/tex]
- [tex]\( 63 = 3^2 \times 7 \)[/tex]
2. Identify the highest power of each prime factor:
- For prime number 2: Highest power is [tex]\( 2^2 \)[/tex]
- For prime number 3: Highest power is [tex]\( 3^3 \)[/tex]
- For prime number 7: Highest power is [tex]\( 7^1 \)[/tex]
3. Calculate the LCM:
- LCM = [tex]\( 2^2 \times 3^3 \times 7^1 \)[/tex]
- LCM = [tex]\( 4 \times 27 \times 7 \)[/tex]
- LCM = [tex]\( 4 \times 189 \)[/tex]
- LCM = [tex]\( 756 \)[/tex]
The least common multiple of 36, 54, and 63 is [tex]\( 756 \)[/tex].
Thus, the best set to use for assessing the student's understanding of finding the LCM of three numbers is Set 3: [tex]\( 36, 54, 63 \)[/tex].
