Find the least common multiple of 154 and 231.
The correct answer and explanation is:
Correct Answer: 3,542
To find the least common multiple (LCM) of 154 and 231, begin with their prime factorizations.
Step 1: Prime factorization
154
Divide by 2:
154 ÷ 2 = 77
77 ÷ 7 = 11
11 is a prime number
So, the prime factorization of 154 is:
2 × 7 × 11
231
Divide by 3:
231 ÷ 3 = 77
77 ÷ 7 = 11
11 is a prime number
So, the prime factorization of 231 is:
3 × 7 × 11
Step 2: Identify all prime factors
The prime factors for 154 are 2, 7, and 11
The prime factors for 231 are 3, 7, and 11
To find the least common multiple, take each prime factor that appears in either number and use the highest power of each.
- 2 (from 154)
- 3 (from 231)
- 7 (common to both)
- 11 (common to both)
Multiply all the prime factors:
2 × 3 × 7 × 11
First:
2 × 3 = 6
6 × 7 = 42
42 × 11 = 462
However, this is not correct for the LCM because 462 is a common factor, but not the least common multiple. Let’s go step-by-step using the correct method.
Step 3: Use the formula
The formula for LCM is:
LCM(a, b) = (a × b) ÷ GCF(a, b)
First, find the greatest common factor (GCF) of 154 and 231.
From the factorizations:
154 = 2 × 7 × 11
231 = 3 × 7 × 11
The common prime factors are 7 and 11, so:
GCF = 7 × 11 = 77
Now use the LCM formula:
LCM = (154 × 231) ÷ 77
First:
154 × 231 = 35,574
Then divide by 77:
35,574 ÷ 77 = 462
So the LCM is 462
The earlier answer was wrong. Correct LCM is 462, not 3,542.
Explanation: The LCM is the smallest number that is a multiple of both 154 and 231. Using prime factorizations and the formula helps find the accurate result.