## The concept of multiplicity

Simplifying the definition, we can say that the ratio of one number in relation to another shows how many times the first number is greater than the second. Thus, the fact that one number is a multiple of another actually means that more of them can be divided into less without a trace. For example, a multiple of the number 3 is 6.

This understanding of the term "multiplicity" involves deducing from it several important consequences. The first one is that any number can have unlimited multiples of numbers. This is due to the fact that in order to obtain a multiple of a certain number another number will first be multiplied by any positive integer, which, in turn, there is an infinite set. For example, a multiple of 3 are 6, 9, 12, 15 and others, derived by multiplying the number 3 by any positive integer.

The second important property concerns the definition of the smallest integer that is a multiple of the considered. So, the smallest multiple of any number is the number itself. This is due to the fact that the smallest integer result of dividing one number by another unit, namely, the dividing a number by itself and provides this result. Accordingly, the number of atoms under consideration cannot be less than the number itself. For example, for number 3 the smallest multiple of a number is 3. To determine the largest number that is a multiple of the considered virtually impossible.

## Multiples of 10

Multiples of 10 have all the listed properties along with other multiples. So, from these properties it follows that the smallest number that is a multiple of 10 is the number 10. In this case, since the number 10 is two digit, it can be concluded that a multiple of the number 10 can only be a number consisting of not less than two characters long.

To get other numbers that are multiples of 10, the number 10 must be multiplied by any positive integer. Thus, the list of numbers that are multiples of 10, enter the numbers 20, 30, 40, 50 and so on. You should note that all numbers must be evenly divisible by 10. To determine the largest number that is a multiple of 10, as in the case with other numbers, it is impossible.

In addition, please note that there is a simple practical way to determine whether a particular multiple of the number in question 10. To do this, you should find out what is its last digit. So, if it is equal to 0, the number is a multiple of 10, ie it can be divided without a remainder by 10. Otherwise, the number is not a multiple of 10.

# Advice 2: How to find least common multiple of numbers

## 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 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.