## Finding least common multiple: basic concepts

To understand how to calculate LCM, should be determined primarily by the value of the term "multiples".

A multiple of the number And call this natural number which evenly divides into A. So, multiples of 5 can be considered 15, 20, 25 and so on.

Of divisors of a particular number can be limited, but a multiple of an infinite set.

Common multiple of natural numbers is a number that divides them without a remainder.

## How to find least common multiple of numbers

Least common multiple (LCM) of numbers (two, three or more) is the smallest natural number that is divisible by all those numbers evenly.

To find the LCM, you can use several ways.

For small numbers it is convenient to write in the line are all multiples of these numbers until for are common. Multiples indicate in the record the capital letter K.

For example, a multiple of the number 4 can be written as:

To (4) = {8,12, 16, 20, 24, ...}

To (6) = {12, 18, 24, ...}

So, you can see that the least common multiple of numbers 4 and 6 is the number 24. This recording is performed as follows:

Knock (4, 6) = 24

If the numbers are large, or need to find the least common multiple of three or more numbers, it is better to use a different method to calculate the NOC.

To perform the job must be expanded the proposed number into Prime factors.

First you need to write in place of the decomposition of most of the numbers and the rest.

In the decomposition of each number may contain a different number of multipliers.

For example, decompose into Prime factors of the number 50 and 20.

50 = 2 * 5 * 5

20 = 2 * 5 * 2

In the decomposition of a smaller number, it should be emphasized multipliers, which are absent in the decomposition of the first of a very large number and then add to it. In the presented example is not enough of two.

You can now calculate the least common multiple of 20 and 50.

NOC (20, 50) = 2 * 5 * 5 * 2 = 100

So, the product of the Prime factors of a larger number of factors of the second numbers, which are not included in the expansion of the greater will be the lowest common multiple.

To find the NOC three numbers or more, they should all be decomposed into Prime factors, as in the previous case.

As an example, it is possible to find the least common multiple of the numbers 16, 24, 36.

36 = 2 * 2 * 3 * 3

24 = 2 * 2 * 2 * 3

16 = 2 * 2 * 2 * 2

So, in the expansion of the greater number of the multipliers does not include only two twos from the decomposition of sixteen (one is in the decay of twenty-four).

Thus, you need to add them to the decomposition of the greater number.

NOC (12, 16, 36) = 2 * 2 * 3 * 3 * 2 * 2 = 9

There are special cases of determining the least common multiple. So, if one of the numbers can be evenly divided by another, the larger of these numbers will be the least common multiple.

For example, the NOC of twelve and twenty-four is twenty-four.

If you want to find the least common multiple of relatively Prime numbers that do not have the same divisors, their NOC will be equal to their product.

For example, LCM (10, 11) = 110.

# Advice 2 : How to find least common divisor

**divisor**of the denominators and calculate.

- - the ability to decompose the number into Prime factors;
- - the ability to perform operations with fractions.

# Advice 3 : How to find gcd and LCM of numbers

a = b*k_0 + r_1

b = r_1*k_1 + r_2

r_1 = r_2*k_2 + r_3

...

r_(n - 1) = r_n*k_n,

where k_i is an integer multiplier.

GCD (a, b) = r_n.

Find GCD (36, 120). The Euclidean algorithm subtract 120, the number that is a multiple of 36, in this case 120 – 36*3 = 12. Now subtract 120, the number of multiples of 12, will 120 – 12*10 = 0. Therefore, GCD (36, 120) = 12.

GCD (a, b) = 2*GCD (a/2, b/2) for odd a and b

GCD (a, b) = GCD (a/2, b) for even a and odd b (the opposite is true GCD (a, b) = GCD (a, b/2))

GCD (a, b) = GCD ((a - b)/2, b) for odd a > b

GCD (a, b) = GCD ((b - a)/2, a) for odd b > a

Thus, GCD (36, 120) = 2*GCD (18, 60) = 4*GCD (9, 30) = 4* GCD (9, 15) = 4*GCD ((15 - 9)/2=3, 9) = 4*3 = 12.

The NOC can be calculated using the GCD: NOC (a, b) = |a*b|/GCD (a, b).

a = r_1^k_1*...*r_n^k_n

b = r_1^m_1*...*r_n^m_n,

where the r_i are all Prime numbers and k_i and m_i are integers ≥ 0.

The NOC is represented in the form of the same Prime factors, where the degree is taken for the maximum of two numbers.

Find NOK (16, 20):

16 = 2^4*3^0*5^0

20 = 2^2*3^0*5^1

NOC (16, 20) = 2^4*3^0*5^1 = 16*5 = 80.