The lcm of 148 and 185 is

Mathematics
1

the lcm of 148 and 185 is

2

What is the LCM of 148 and 185?

Answer:

To find the least common multiple (LCM) of 148 and 185, we employ the prime factorization method to ensure a comprehensive understanding of the underlying principles. This method involves finding the prime factors of each number and then using these to determine the LCM.

Step 1: Prime Factorization

Prime Factorization of 148:

  1. Divide by the smallest prime number: Start with 2, the smallest prime number.
    [
    148 \div 2 = 74
    ]
  2. Continue dividing:
    [
    74 \div 2 = 37
    ]
  3. Stop when reaching a prime number: The number 37 is a prime number.

Thus, the prime factorization of 148 is 2^2 \times 37.

Prime Factorization of 185:

  1. Divide by the smallest prime number: Test divisibility by 2 (it isn’t even) and 3 (the sum of digits isn’t divisible by 3). Move to 5. The number 185 ends in 5, so it’s divisible by 5.
    [
    185 \div 5 = 37
    ]
  2. Stop when reaching a prime number: The number 37 is the remaining number and is prime.

Thus, the prime factorization of 185 is 5 \times 37.

Step 2: Determine the LCM

To find the LCM, take the highest power of each prime factor involved across both numbers.

  • Prime factor 2: Maximum power = (2^2) (from 148)
  • Prime factor 5: Maximum power = (5^1) (from 185)
  • Prime factor 37: Maximum power = (37^1) (common in both)

The LCM is obtained by multiplying these factors together:

LCM = 2^2 \times 5^1 \times 37^1

Calculation:

[
LCM = 4 \times 5 \times 37
]

  1. Multiply (4 \times 5):
    [
    4 \times 5 = 20
    ]

  2. Multiply (20 \times 37):
    [
    20 \times 37 = 740
    ]

Thus, the LCM of 148 and 185 is 740.

Final Answer:

The LCM of 148 and 185 is 740.