You will need

- - a table of Prime numbers;
- - divisibility rules of numbers;
- calculator.

Instruction

1

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".

2

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.

3

For example, the number of common factors of a number find 364 thus:

364/2=182

182/2=91

91/7=13

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).

364/2=182

182/2=91

91/7=13

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).

4

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.

5

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)∙...

6

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.You will need

- Divisibility rules

Instruction

1

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).

2

Numbers that have more than two divisors are called composite. What

Since even

**number**can be composite?Since even

**numbers**are divisible by 2 divisible, then all the even**numbers**except**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.3

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**.4

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**24.5

However, not all even numbers, this scheme can give all

There are divisibility rules for certain

Divisibility by 3: when the sum of the digits

Divisibility by 5: last digit when

Divisibility by 7: when the result of subtracting twice the last digit from this

Divisibility by 9 when the sum of digits

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

**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**.6

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).