The least common multiple (MMC or M.M.C) and the greatest common divisor (MDC or M.D.C) can be calculated simultaneously by decomposing into prime factors.
Through factorization, the MMC of two or more numbers is determined by multiplying the factors. The MDC, on the other hand, is obtained by multiplying the numbers that divide them at the same time.
1st step: factoring the numbers
Factorization consists of representing prime numbers, which are called factors. For example, 2 x 2 is the factored shape of 4.
The factored form of a number is obtained by following the sequence:
- It starts with division by the smallest possible prime number;
- The quotient of the previous division is also divided by the smallest possible prime number;
- The division is repeated until the result is number 1.
Example: factoring the number 40.
40 | 2 → 40: 2 = 20, since 2 is the smallest possible prime divisor and the division quotient is 20.
20 | 2 → 20: 2 = 10, since 2 is the smallest possible prime divisor and the division quotient is 10.
10 | 2 → 10: 2 = 5, since 5 is the smallest possible prime divisor and the division quotient is 5.
5 | 5 → 5: 5 = 1, since 5 is the smallest possible prime divisor and the division quotient is 1.
1
So the factored form of the number 40 is 2 x 2 x 2 x 5, which is the same as 23 x 5.
Learn more about Prime numbers.
2nd step: MMC calculation
Decomposing two numbers simultaneously will result in the factored form of the least common multiple between them.
Example: factoring the numbers 40 and 60.
The multiplication of prime factors 2 x 2 x 2 x 3 x 5 has a factored form 23 x 3 x 5.
Therefore, the MMC of 40 and 60 is: 23 x 3 x 5 = 120.
Remember that the divisions will always be done by the smallest possible prime number, even if this number only divides one of the components.
Learn more about Least common multiple.
3rd step: MDC calculation
The greatest common divisor is found when we multiply the factors that simultaneously divide the factored numbers.
In factoring 40 and 60, we can see that number 2 was able to divide the division quotient twice and number 5 once.
Therefore, the MDC of 40 and 60 is: 22 x 5 = 20.
Learn more aboutMaximum Common Divider.
Practicing MMC and MDC calculations
Exercise 1: 10, 20 and 30
Correct answer: MMC = 60 and MDC = 10.
1st step: decomposition into prime factors.
Divide by the smallest possible prime numbers.
2nd step: MMC calculation.
Multiply the factors found above.
MMC: 2 x 2 x 3 x 5 = 22 x 3 x 5 = 60
3rd step: calculation of the MDC.
Multiply the factors that divide the numbers at the same time.
MDC: 2 x 5 = 10
Exercise 2: 15, 25 and 45
Correct answer: MMC = 225 and MDC = 5.
1st step: decomposition into prime factors.
Divide by the smallest possible prime numbers.
2nd step: MMC calculation.
Multiply the factors found above.
MMC: 3 x 3 x 5 x 5 = 32 x 52 = 225
3rd step: MDC calculation
Multiply the factors that divide the numbers at the same time.
MDC: 5
See too: Multiples and Dividers
Exercise 3: 40, 60 and 80
Correct answer: MMC = 240 and MDC = 20.
1st step: decomposition into prime factors.
Divide by the smallest possible prime numbers.
2nd step: MMC calculation.
Multiply the factors found above.
MMC: 2 x 2 x 2 x 2 x 3 x 5 = 24 x 3 x 5 = 240
3rd step: calculation of the MDC.
Multiply the factors that divide the numbers at the same time.
MDC: 2 x 2 x 5 = 22 x 5 = 20
For more issues with commented resolution, see also: MMC and MDC - Exercises.
