The LCM (98,713,252 and 23,521) = ? 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 least common multiple
lcm (98,713,252; 23,521) = ?
Method 1. The prime factorization:
The prime factorization of a number: finding the prime numbers that multiply together to make that number.
98,713,252 = 22 × 112 × 203,953
98,713,252 is not a prime number but a composite one.
23,521 = 43 × 547
23,521 is not a prime number but a composite one.
* Prime number: a natural number that is only divisible by 1 and itself. A prime number has exactly two factors: 1 and itself.
* Composite number: 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 (98,713,252; 23,521) = 22 × 112 × 43 × 547 × 203,953 = 2,321,834,400,292
The two numbers have no prime factors in common
2,321,834,400,292 = 98,713,252 × 23,521
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:
98,713,252 ÷ 23,521 = 4,196 + 19,136
Step 2. Divide the smaller number by the above operation's remainder:
23,521 ÷ 19,136 = 1 + 4,385
Step 3. Divide the remainder of the step 1 by the remainder of the step 2:
19,136 ÷ 4,385 = 4 + 1,596
Step 4. Divide the remainder of the step 2 by the remainder of the step 3:
4,385 ÷ 1,596 = 2 + 1,193
Step 5. Divide the remainder of the step 3 by the remainder of the step 4:
1,596 ÷ 1,193 = 1 + 403
Step 6. Divide the remainder of the step 4 by the remainder of the step 5:
1,193 ÷ 403 = 2 + 387
Step 7. Divide the remainder of the step 5 by the remainder of the step 6:
403 ÷ 387 = 1 + 16
Step 8. Divide the remainder of the step 6 by the remainder of the step 7:
387 ÷ 16 = 24 + 3
Step 9. Divide the remainder of the step 7 by the remainder of the step 8:
16 ÷ 3 = 5 + 1
Step 10. Divide the remainder of the step 8 by the remainder of the step 9:
3 ÷ 1 = 3 + 0
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 (98,713,252; 23,521) = 1
2. Calculate the least common multiple:
The least common multiple, Formula:
lcm (a; b) = (a × b) / gcf, hcf, gcd (a; b)
lcm (98,713,252; 23,521) =
(98,713,252 × 23,521) / gcf, hcf, gcd (98,713,252; 23,521) =
2,321,834,400,292 / 1 =
2,321,834,400,292
The least common multiple:
lcm (98,713,252; 23,521) = 2,321,834,400,292 = 22 × 112 × 43 × 547 × 203,953
Why is it useful to calculate the least common multiple?
In order to add, subtract or sort fractions with different denominators, we must make their denominators the same. An easy way is to calculate the least common multiple of all the denominators (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:
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.