The LCM (7,291 and 84) = ? Calculate the Least (the Lowest) Common Multiple, LCM, by Using Two Methods: 1) The Prime Factorization of the Numbers and 2) The Euclidean Algorithm

The prime factorization of a number: finding the prime numbers that multiply together to make that number.

7,291 = 23 × 317
7,291 is not a prime number but a composite one.

84 = 2² × 3 × 7
84 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.

>> The prime factorization

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).

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:
7,291 ÷ 84 = 86 + 67

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Step 2. Divide the smaller number by the above operation's remainder:
84 ÷ 67 = 1 + 17

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Step 3. Divide the remainder of the step 1 by the remainder of the step 2:
67 ÷ 17 = 3 + 16

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Step 4. Divide the remainder of the step 2 by the remainder of the step 3:
17 ÷ 16 = 1 + 1

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Step 5. Divide the remainder of the step 3 by the remainder of the step 4:
16 ÷ 1 = 16 + 0

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At this step, the remainder is zero, so we stop:
1 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 (7,291; 84) = 1

The least common multiple, Formula:

lcm (a; b) = ^{(a × b)} / _{gcf, hcf, gcd (a; b)}

>> The Euclidean Algorithm

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.