The calculations of MMC and MDC are related to multiples and divisors of a natural number. By multiple we mean the product generated by the multiplication between two numbers.
Watch:
We say that 30 is a multiple of 5, since 5·6 = 30. There is a natural number that multiplied by 5 results in 30. See some more numbers and their multiples:
M(3) = 0, 3, 6, 9, 12, 15, 18, 21, …
M(4) = 0, 4, 8, 12, 16, 20, 24, 28, 32, …
M(10) = 0, 10, 20, 30, 40, 50, 60, …
M(8) = 0, 8, 16, 24, 32, 40, 48, 56, …
M(20) = 0, 20, 40, 60, 80, 100, 120, …
M(11) = 0, 11, 22, 33, 44, 55, 66, 77, 88, 99, ...
You multiples of a number form an infinite set of elements.
dividers
One number is considered divisible by another when the remainder of the division between them is equal to zero. Note some numbers and their divisors:
D(10) = 1, 2, 5, 10.
D(20) = 1, 2, 4, 5, 10, 20.
D(25) = 1, 5, 25.
D(100) = 1, 2, 4, 5, 10, 20, 25, 50, 100.
Minimum Common Multiple (MMC)
O least common multiple between two numbers is represented by the smallest common value belonging to the multiples of the numbers. Note the MMC between numbers 20 and 30:
M(20) = 0, 20, 40, 60, 80, 100, 120, ...
M(30) = 0, 30, 60, 90, 120, 150, 180, …
MMC between 20 and 30 is equivalent to 60.
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Another way to determine the MMC between 20 and 30 is through factorization, in which we must choose common and non-common factors with the greatest exponent. Watch:
20 = 2·2·5 = 2²·5
30 = 2·3·5 = 2·3·5
MMC (20, 30) = 2²·3·5 = 60
The third option is to perform the simultaneous decomposition of numbers, multiplying the factors obtained. Watch:
20, 30| 2 10, 15| 2 5, 15| 3 5, 5| 5 1, 1|
MMC(20.30) = 2·2·3·5 = 60
Maximum Common Divider (MDC)
The greatest common divisor between two numbers is represented by the greatest common value belonging to the number's divisors. Note the MDC between numbers 20 and 30:
D(20) = 1, 2, 4, 5, 10, 20.
D(30) = 1, 2, 3, 5, 6, 10, 15, 30.
The greatest common divisor of the numbers 20 and 30 is 10.
We can also determine the MDC between two numbers through factorization, in which we choose the common factors with the smallest exponent. Note the MDC of 20 and 30 from this method.
20 = 2·2·5 = 2²·5
30 = 2·3·5 = 2·3·5
MDC (20, 30) = 2·5 = 10
Example:
Let's determine the MMC and MDC between the numbers 80 and 120.
MMC
80 = 2·2·2·2·5 = 24·5
120 = 2·2·2·3·5 = 2³·3·5
MMC (80, 120) = 24·3·5 = 240
MDC (80, 120) = 2³·5 = 40
by Mark Noah
Graduated in Mathematics
