Advice 1: How to find the number of divisors

In the most General case, the number of possible divisors of an arbitrary number of infinitely. In fact, it's not equal to zero. But if we are talking about natural numbers, then the divisor of N means a is a natural number that evenly divides a number N. the Number of such divisors is always limited, and you can find them with the help of special algorithms. There are also Prime factors of the numbers that represent Prime numbers.
How to find the number of divisors
You will need
  • - a table of Prime numbers;
  • - divisibility rules of numbers;
  • calculator.
Most often, it is necessary to decompose the number into Prime factors. Are numbers that divide the original number without a remainder, and can divide evenly only by itself and the unit (such numbers include 2, 3, 5, 7, 11, 13, 17 etc.). Moreover, no regularity in the number of Prime numbers not found. Take them from a special table or search using the algorithm called "sieve of Eratosthenes".
Start to select a Prime number that divides the given number. Private again divide by a Prime number and continue this process until, as long as private will not remain a Prime number. Then just count the number of Prime factors, add to it the number 1 (which takes into account the last private). The result will be the number of Prime factors that when multiplied will give the required number.
For example, the number of common factors of a number find 364 thus:


Get numbers 2, 2, 7, 13, which are simple natural divisors of the number 364. Their number is equal to 3 (if you count the duplicate splitters in one).
If you need to find the total number of all possible positive divisors of a number, use its canonical decomposition. To do this, as described above decompose the number into Prime factors. Then record the number as the product of these multipliers. Repeating the number raise to the extent, for example, if you have received three divider 5, then write it down as 53.
Write down the work from the smallest to the largest multipliers. Such a work and called the canonical decomposition of the number. Each multiplier of this decomposition has a degree, is represented by natural number (1, 2, 3, 4, etc.). Denote the exponents of the multipliers A1, A2, A3, etc. Then the total number of divisors will be equal to the product (a1 + 1)∙(a2 + 1)∙(a3+1)∙...
For example, take the same number 364: its canonical decomposition 364=22∙7∙13. Get A1=2, A2=1, A3=1, then the number of positive divisors of this number will be equal to (2+1)∙(1+1)∙(1+1)=3∙2∙2=12.

Advice 2 : How to find all factors of a number

The number b is called the divisor a whole number a, if there is an integer q such that bq = a. Usually considered the divisibility of natural numbers. The dividend itself will be called a multiple of the number b. Search all divisors of the number is carried out according to certain rules.
How to find all factors of a number
You will need
  • Divisibility rules
First, make sure that any natural number greater than one, has at least two divisors - one and itself. Indeed, a:1 = a, a:a = 1. Numbers that have only two divisors, are called simple. The only divider unit is obviously the unit. That is, one is not a Prime number (and is not composite, as we will see later).
Numbers that have more than two divisors are called composite. What number can be composite?
Since even numbers are divisible by 2 divisible, then all the even numbersexcept the number 2, are composite. Indeed, when dividing 2:2 deuce divided within itself, ie it has only two divisor (1 and 2) and is a simple number.
Will see if there are an even number of even some dividers. Divide it first by 2. From the commutative operation of multiplication it is obvious that the resulting quotient will also be a divisor of the number. Then, if the resulting quotient is an integer, and divide again by 2 already it's private. Then the resulting new quotient y = (x:2):2 = x:4 will also be a divisor of the original numbers. Similarly, 4 is a divisor of the original numbers.
Continuing this chain, we can generalize the rule: sequentially divide an even number first and then the resulting private for 2 to until a certain individual will not be equal to an odd number. In this case, all the resulting private are divisors of this number. In addition, the divisors of this number will be the number 2^k where k = 1...n, where n is the number of steps of the chain.Example: 24:2 = 12, 12:2 = 6, 6:2 = 3 - an odd number. Consequently, the 12, 6 and 3 factors of the number 24. In the chain 3 steps, therefore, divisors of the number 24 will also be the number 2^1 = 2 (already known from the parity of the number 24), 2^2 = 4 2^3 = 8. Thus, the number of 1, 2, 3, 4, 6, 8, 12 and 24 are divisors of the number 24.
However, not all even numbers, this scheme can give all the divisors of the number. Consider, for example, the number 42. 42:2 = 21. However, as we know, the numbers 3, 6 and 7 will also be divisors of the number 42.
There are divisibility rules for certain numbers. Will discuss the most important of them:
Divisibility by 3: when the sum of the digits of the number is divisible by 3 without a remainder.
Divisibility by 5: last digit when the number is 5 or 0.
Divisibility by 7: when the result of subtracting twice the last digit from this number the last number is divided by 7.
Divisibility by 9 when the sum of digits of number divisible by 9 without a remainder.
Divisibility by 11: when the sum of digits occupying odd places, or equal to the sum of digits occupying odd places, or differ from it by a number divisible by 11.
There are also divisibility rules for 13, 17, 19, 23 and other numbers.
As for even and for odd numbers you need to use signs division for a particular number. Dividing a number, you should determine the divisors of the resulting private, etc. (the chain is the same chain of even numbers by dividing them by 2, described above).

Advice 3 : 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 4 : 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.
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