The least common multiple (lcm). What it is and how to calculate it.
- The number 60 is a common multiple of the numbers 6 and 15 because 60 is a multiple of 6 (60 = 6 × 10) and also a multiple of 15 (60 = 15 × 4).
- There are infinitely many common multiples of 6 and 15.
- If the number "v" is a multiple of the numbers "a" and "b", then all the multiples of "v" are also multiples of "a" and "b".
- The common multiples of 6 and 15 are the numbers 30, 60, 90, 120, and so on.
- Out of these, 30 is the smallest, 30 is the least common multiple (lcm) of 6 and 15.
- Note: The prime factorization of a number: finding the prime numbers that multiply together to give that number.
- If e = lcm (a, b), then the prime factorization of "e" must contain all the prime factors involved in the prime factorization of "a" and "b" taken by the highest power.
- Example:
- 40 = 23 × 5
- 36 = 22 × 32
- 126 = 2 × 32 × 7
- lcm (40, 36, 126) = 23 × 32 × 5 × 7 = 2,520
- Note: 23 = 2 × 2 × 2 = 8. We are saying that 2 was raised to the power of 3. Or, shorter, 2 to the power of 3. In this example 3 is the exponent and 2 is the base. The exponent indicates how many times the base is multiplied by itself. 23 is the power and 8 is the value of the power:
- Another example of calculating the least common multiple, lcm:
- 938 = 2 × 7 × 67
- 982 = 2 × 491
- 743 = is a prime number and cannot be broken down into other prime factors
- lcm (938, 982, 743) = 2 × 7 × 67 × 491 × 743 = 342,194,594
- If two or more numbers have no common factors (they are coprime), then their least common multiple is calculated by simply multiplying the numbers.
- Example:
- 6 = 2 × 3
- 35 = 5 × 7
- lcm (6, 35) = 2 × 3 × 5 × 7 = 6 × 35 = 210