Advice 1: How to find least common multiple of numbers

Students often meet among the her math homework the following formulation: "find the least common multiple of numbers". This definitely need to learn to do in order to perform various operations with fractions with different denominators.
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 2 : How to find least common divisor

Addition and subtraction of natural fractions only when they have the same denominator. In order to simplify the calculations when casting their common denominator, find lowest common divisor of the denominators and calculate.
How to find least common divisor
You will need
  • - the ability to decompose the number into Prime factors;
  • - the ability to perform operations with fractions.
Write down the mathematical operation for addition of fractions. Then, find their least common multiple. To do this, perform the following steps: 1. Imagine each of the denominators as a product of Prime numbers (a Prime number is a number that is divisible without a remainder only by 1 and itself, e.g. 2, 3, 5, 7, etc.).2. Group all simple dividers that are drawn by specifying their degree. 3. Select the highest degree of each of these Prime factors that occur in these numbers. 4. Multiply the prescribed degree.
For example, the common denominator for fractions with denominators of 15, 24 and 36 is the number that plan: 15=3•5; 24=2^3•3;36=2^3•3^2.Enter the greatest of all Prime divisors of these numbers: 2^3•3^2•5=360.
Divide the common denominator for each and add up the denominators of the fractions. On the number you multiply their numerators. Under the General feature of fractions write the least common numerator, which is both the lowest common denominator. In the numerator, add the numbers obtained by multiplying each numerator into the quotient of the least common of the dividend by the denominator of the fraction. The sum of all numerators divided by the least common denominator will be sought.
For example, to adding fractions 4/15, 7/24 and 11/36 do so. Find the least common denominator, which is 360. Then we split 360/15=24, 360/24=15, 360/36=10. The number 4, which is the numerator of the first fraction, multiply by 24 (4•24=96), number 7 15 (7•15=105), 11 10 (11•10=110). Then add up these numbers (96+105+110=301). Get the result 4/15+7/24+11/36=301/360.

Advice 3 : How to find gcd and LCM of numbers

Integers – the set of mathematical integers, which is of great use in everyday life. Nonnegative integers are used when specifying the number of any objects, a negative number in messages about the weather forecast and so NOD and NOC are natural features of integers connected with the operations division.
How to find gcd and LCM of numbers
The greatest common divisor (GCD) of two integers is the largest integer that divided both the original numbers without a remainder. At least one of them must be different from zero, and the GCD.
The GCD is easy to calculate the Euclidean algorithm or the binary method. The Euclidean algorithm determine the GCD of integers a and b, one of which is not zero, there is a sequence of numbers r_1 > r_2 > r_3 > ... > r_n, where the element r_1 is equal to the remainder from dividing the first number by the second. And other members of the sequence equals the remnants of the division predprinyavshego member for the previous and the penultimate element is divided into the past without a trace.
Mathematically the sequence can be represented as:
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.
The Euclidean algorithm called mutual subtraction, since the GCD is obtained by successive subtraction of the smaller from the larger. It is easy to assume that GCD (a, b) = GCD (b, r).
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.
Binary the algorithm for finding the GCD is based on the theory of shift. According to this method the GCD of two numbers has the following properties:
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.
Least common multiple (LCM) of two integers is the smallest integer that is a multiple of both original numbers without a remainder.
The NOC can be calculated using the GCD: NOC (a, b) = |a*b|/GCD (a, b).
The second method of calculating the knock – canonical decomposition of numbers into Prime factors:
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.
There is the concept of mutually-Prime numbers have no common divisors except 1. For such numbers GCD (a, b) = 1.
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