LCM (6,513; 52) = ? Calculate the least common multiple, LCM, by two methods: 1) The prime factorization of the numbers and 2) The Euclidean algorithm
lcm (6,513; 52) = ?
Method 1. The prime factorization:
The prime factorization of a number: finding the prime numbers that multiply together to make that number.
6,513 = 3 × 13 × 167
6,513 is not a prime number but a composite one.
52 = 22 × 13
52 is not a prime number but a composite one.
* The natural numbers that are only divisible by 1 and themselves are called prime numbers. A prime number has exactly two factors: 1 and itself.
* A composite number is a natural number that has at least one other factor than 1 and itself.
Calculate the least common multiple, lcm:
Multiply all the prime factors of the two numbers. If there are common prime factors then only the ones with the largest exponents are taken (the largest powers).
The least common multiple:
lcm (6,513; 52) = 22 × 3 × 13 × 167 = 26,052
Method 2. The Euclidean Algorithm:
1. Calculate the greatest (highest) common factor (divisor):
This algorithm involves the process of dividing numbers and calculating the remainders.
'a' and 'b' are the two natural numbers, 'a' >= 'b'.
Divide 'a' by 'b' and get the remainder of the operation, 'r'.
If 'r' = 0, STOP. 'b' = the gcf (hcf, gcd) of 'a' and 'b'.
Else: Replace ('a' by 'b') and ('b' by 'r'). Return to the step above.
Step 1. Divide the larger number by the smaller one:
6,513 ÷ 52 = 125 + 13
Step 2. Divide the smaller number by the above operation's remainder:
52 ÷ 13 = 4 + 0
At this step, the remainder is zero, so we stop:
13 is the number we were looking for - the last non-zero remainder.
This is the greatest (highest) common factor (divisor).
The greatest (highest) common factor (divisor):
gcf, hcf, gcd (6,513; 52) = 13
2. Calculate the least common multiple:
The least common multiple, Formula:
lcm (a; b) = (a × b) / gcf, hcf, gcd (a; b)
lcm (6,513; 52) =
(6,513 × 52) / gcf, hcf, gcd (6,513; 52) =
338,676 / 13 =
26,052
The least common multiple:
lcm (6,513; 52) = 26,052 = 22 × 3 × 13 × 167
The two numbers have common prime factors
Why is it useful to calculate the least common multiple?
When adding, subtracting or sorting fractions with different denominators, in order to work with those fractions we must first make the denominators the same. An easy way is to calculate the least common multiple of all the denominators of the fractions (the least common denominator).
By definition, the least common multiple of two numbers is the smallest natural number that is: (1) greater than 0 and (2) a multiple of both numbers.
Other similar operations with the least common multiple:
The least common multiple, LCM: the latest 5 calculated values
Calculator: calculate the least common multiple, lcm
Calculate the least common multiple of the numbers, LCM:
Method 1: Run the prime factorization of the numbers - then multiply all the prime factors of the numbers, taken by the largest exponents.
Method 2: The Euclidean algorithm:
lcm (a; b) = (a × b) / gcf (a; b)
Method 3: The divisibility of the numbers.
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
Some articles on the prime numbers