# Least Common Multiple (M.M.C.)

## Multiple of a natural number

How 24 is divisible by 3, we say that 24 is a multiple of 3.

24 is also a multiple of 1, 2, 3, 4, 6, 8, 12, and 24.

If one number is divisible by another, nonzero, so
we say he is multiple this other one.

Multiples of a number are calculated by multiplying that number by the natural numbers.

Example: The multiples of 7 are:
7x0, 7x1, 7x2, 7x3, 7x4,… = 0, 7, 14, 21, 28,…

Important Notes:
1) A number has multiple infinities
2) Zero is multiple of any natural number

## What is M.M.C.?

Two or more numbers always have multiples common to them. Let's find the common multiples of 4 and 6:
Multiples of 6: 0, 6, 12, 18, 24, 30,…
Multiples of 4: 0, 4, 8, 12, 16, 20, 24,…
Common multiples of 4 and 6: 0, 12, 24,…

Among these nonzero multiples, 12 is the smallest of them. We call the 12 least common multiple of 4 and 6.

The smallest common multiple of two or more nonzero numbers is called the least common multiple of these numbers. We use the abbreviation m.m.c.

## M.M.C. Calculation

We can calculate m.m.c. of two or more numbers using factorization. Follow the calculation of m.m.c. 12 and 30:

1º) we break down the numbers into prime factors
2º) the m.m.c. is the product of common and non-common prime factors:

12 = 2x2x3
30 = 2x3x5
m.m.c (12.30) = 2 x 2 x 3 x 5

Writing the factorization of numbers in the form of power, we have:
12 = 22 x 3
30 = 2x3x5
m.m.c (12.30) = 22 x 3 x 5

O m.m.c. of two or more numbers, when factored, is the product of factors common to them and not common to them, each raised to the greatest exponent.

### Simultaneous decomposition process

In this process, we break down all the numbers at the same time on a device as shown in the next figure. The product of the prime factors we get in this decomposition is m.m.c. of these numbers. The following is the calculation of m.m.c. (15,24,60).

Therefore, m.m.c. (15.24,60) = 2 x 2 x 2 x 3 x 5 = 120

## Property of M.M.C.

Between the numbers 3, 6 and 30, the number 30 is a multiple of the other two. In this case, 30 is m.m.c. (3,6,30). Watch:

m.m.c. (3.6.30) = 2 x 3 x 5 = 30

Given two or more numbers, if one of them is multiple of all the others, then he is m.m.c. of the given numbers.

Consider the numbers 4 and 15, which are prime to each other. The m.m.c. (4.15) is 60, which is the product of 4 by 15. Note:

m.m.c. (4.15) = 2 x 2 x 3 x 5 = 60

Dice Two prime numbers among themselves, O m.m.c. of them is the product of these numbers. Next: Equations of the 1st degree with one variable