Mathematical expressions and problems require a lot of additional knowledge. LCM 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 the material, it will not be difficult for a person familiar with degrees and the multiplication table to select the necessary numbers and find the result.
Definition
Common multiple - a number that can be completely divided 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 divisible by both numbers at once, without deviations.
NOK is the accepted short name for designation, assembled from the first letters.
Ways to get a number
To find the LCM, the method of multiplying numbers is not always suitable, it is much better suited for simple one-digit or two-digit numbers. It is customary to divide large numbers into factors, the larger the number, the moremultipliers will be.
Example 1
For the simplest example, schools usually take simple, one-digit or two-digit numbers. For example, you need 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, there is simply no smaller number.
Example 2
The second version of the task is much more difficult. The numbers 300 and 1260 are given, finding the NOC is mandatory. To solve the task, the following actions are assumed:
Decomposition of the first and second numbers into the simplest factors. 300=22 352; 1260=22 32 5 7. The first stage is completed.
The second stage involves working with the data already received. Each of the received numbers must participate in the calculation of the final result. For each factor, the largest number of occurrences is taken from the original numbers. LCM is a common number, so the factors from the numbers must be repeated in it to the last, even those that are present in one copy. Both initial numbers have in their composition the numbers 2, 3 and 5, in different powers, 7 is only in one case.
To calculate the final result, you need to take each number in the largest of their represented powers, into the equation. It remains only to multiply and get the answer, with the correct filling, the task fits into two steps without explanation:
1) 300=22 352; 1260=22 32 5 7.
2) NOK=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 3001260=378,000.
Check:
6300 / 300=21 is correct;
6300 / 1260=5 is correct.
The correctness of the result is determined by checking - dividing the LCM by both original numbers, if the number is an integer in both cases, then the answer is correct.
What does LCM mean in math
As you know, there is not a single useless function in mathematics, this one 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 of high school. It is also additionally a common divisor for all multiples, if such conditions are in the problem. Such an expression can find a multiple not only of two numbers, but also of 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 common LCM:
1) 250=2510=52 52=53 2 - this example describes in detail factorization, no reduction.
2) 600=6010=323 52;
3) 1500=15100=3353 22;
In order to make an expression, you need to mention all the factors, in this case 2, 5, 3 are given, - for allof these numbers it is required to determine the maximum degree.
NOC=3000
Attention: all factors must be brought to full simplification, if possible, decomposing to the level of single digits.
Check:
1) 3000 / 250=12 is correct;
2) 3000 / 600=5 is correct;
3) 3000 / 1500=2 is correct.
This method does not require any tricks or genius level abilities, everything is simple and straightforward.
One more way
In mathematics, many things are connected, many things can be solved in two or more ways, the same goes for finding the least common multiple, LCM. The following method can be used in the case of simple two-digit and single-digit numbers. A table is compiled in which the multiplier is entered vertically, 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, a number is taken and the results of multiplying this number by integers are written in a row, 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, you need to find the LCM connecting 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 the numbers are quite different, the only common number among them is 210, so it will be the LCM. Among those associated with this calculationprocesses, there is also a greatest common divisor, which is calculated according to similar principles and is often found in neighboring problems. The difference is small, but significant enough, LCM involves calculating a number that is divisible by all given initial values, and GCD involves calculating the largest value by which the original numbers are divisible.
