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 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.

Is the advice useful?