Mathematical expressions and problems require a lot of additional knowledge. NOC is one of the main ones, especially often used in working with fractions. The topic is studied in high school, while it is not particularly difficult to understand material, it is not difficult for a person familiar with the degrees and multiplication table to select the necessary numbers and find the result.
Definition
The common multiple is a number that can completely divide into two numbers at the same time (a and b). Most often, this number is obtained by multiplying the original numbers a and b. The number must be divided immediately into both numbers, without deviations.
NOC is a short name accepted for designation, collected from the first letters.
Ways to get a number
The method of multiplying numbers is not always suitable for finding NOCs; it is much better suited for simple single or double digit numbers. Large numbers are usually divided into factors, the larger the number, the more factors there will be.
Example No. 1
For the simplest example, schools usually use simple, single, or double-digit numbers. For example, it is necessary to solve the following task, find the least common multiple of the numbers 7 and 3, the solution is quite simple, just multiply them. As a result, there is the number 21, a smaller number is simply not.
Example No. 2
The second option is much more complicated. The numbers 300 and 1260 are given; finding the NOC is mandatory. The following actions are supposed to solve the task:
Decomposition of the first and second numbers into simple factors. 300 = 2 2 * 3 * 5 2 ; 1260 = 2 2 * 3 2 * 5 * 7. The first stage is completed.
The second stage involves working with already received data. Each of the numbers received is required to participate in the calculation of the final result. For each factor, the largest number of occurrences is taken from the composition of the original numbers. NOC is a total number, therefore, factors from numbers should be repeated in it all to a single one, even those that are present in one instance. Both initial numbers have in their composition the numbers 2, 3 and 5, in different degrees, 7 is in only one case.
To calculate the final result, it is necessary to take each number in the highest degree presented, in the equation. It remains only to multiply and get the answer, with proper completion the task fits into two actions without explanation:
1) 300 = 2 2 * 3 * 5 2 ; 1260 = 2 2 * 3 2 * 5 * 7.
2) NOC = 6300.
That's the whole problem, if you try to calculate the desired number by multiplying, then the answer will definitely not be correct, since 300 * 1260 = 378,000.
Verification:
6300/300 = 21 - true;
6300/1260 = 5 - true.
The correctness of the result is determined by checking - dividing the NOC by both initial numbers, if the number is integer in both cases, then the answer is correct.
What does NOC mean in mathematics?
As you know, in mathematics there is not a single useless function, this is no exception. The most common purpose of this number is to bring fractions to a common denominator. What is usually studied in grades 5-6. It is also additionally a common divisor for all multiple numbers, if such conditions are in the problem. A similar expression can find a multiple not only to two numbers, but also to a much larger number - three, five, and so on. The more numbers, the more actions in the task, but the complexity of this does not increase.
For example, given the numbers 250, 600 and 1500, you need to find their total NOC:
1) 250 = 25 * 10 = 5 2 * 5 * 2 = 5 3 * 2 - this example describes in detail the factorization, without reduction.
2) 600 = 60 * 10 = 3 * 2 3 * 5 2 ;
3) 1500 = 15 * 100 = 33 * 5 3 * 2 2 ;
In order to compose an expression, it is necessary to mention all the factors, in this case 2, 5, 3 are given - for all these numbers it is necessary to determine the maximum degree.
NOC = 3000
Attention: all factors must be brought to complete simplification, if possible, expanding to the level of single-valued ones.
Verification:
1) 3000/250 = 12 - true;
2) 3000/600 = 5 - true;
3) 3000/1500 = 2 - true.
This method does not require any tricks or abilities of the genius level, everything is simple and clear.
Another way
In mathematics, much is connected, much can be solved in two or more ways, the same goes for the search for the least common multiple, NOC. The following method can be used in the case of simple two-digit and single-digit numbers. A table is compiled, in which the multiplied vertically is entered, the multiplier horizontally, and the product is indicated in the intersecting cells of the column. You can reflect the table by means of a line, take a number and write in a row the results of multiplying this number by integers, from 1 to infinity, sometimes 3-5 points are enough, the second and subsequent numbers are subjected to the same computational process. Everything happens until a common multiple is found.
Task.
Given the numbers 30, 35, 42, it is necessary to find the NOC linking all the numbers:
1) Multiples of 30: 60, 90, 120, 150, 180, 210, 250, etc.
2) Multiples of 35: 70, 105, 140, 175, 210, 245, etc.
3) Multiples of 42: 84, 126, 168, 210, 252, etc.
It is noticeable that all numbers are quite different, the only common number among them is 210, so it will be the NOC. Among the processes associated with this computation, there is also the largest common divisor, calculated by similar principles and often found in neighboring problems. The difference is small, but significant enough, the NOC assumes the calculation of a number that is divided by all the original values, and the GCD assumes the calculation of the largest value by which the original numbers are divided.