The least common multiple of the NOC

In order to find a common denominator when adding and subtracting fractions with different denominators, it is necessary to know and be able to calculate the least common multiple (LCM).

The multiple of a is the number itself divisible by the number a without a remainder.

Numbers are multiples of 8 (that is, these numbers are divided by 8 without a residue): these are numbers 16, 24, 32 ...

The multiples of 9: 18, 27, 36, 45 ...

The numbers that are multiples of a given number a are infinitely many, in contrast to the divisors of the same number. Dividers - a finite amount.

A common multiple of two natural numbers is a number that is divided into both these numbers completely .

How to find the NOC

NOC can be found and recorded in two ways.

The first way to find the NOC

This method is usually used for small numbers.

1. We write in the line multiple for each of the numbers, until there is a multiple, the same for both numbers.
2. The multiple of a is denoted by a capital “K”.
   
   K (a) = {..., ...}

Example. Find the NOC 6 and 8.

The second way to find the NOC

This method is convenient to use to find the LCM for three or more numbers.

1. Spread these numbers into prime factors. You can read more about the rules for decomposing into simple factors in the topic on how to find the greatest common divisor (GCD).
2. Write in a line the factors included in the expansion of the largest of the numbers, and below it - the decomposition of the remaining numbers.
   
   

   The number of identical factors in the decomposition of numbers can be different.

   60 = 2 • 2 • 3 • 5
   
   24 = 2 • 2 • 2 • 3
   
   
3. Underline the expansion of a smaller number (smaller numbers) of factors that are not included in the expansion of a larger number (in our example, this is 2) and add these factors to the expansion of a larger number.
   LCM (24, 60) = 2 • 2 • 3 • 5 • 2
4. Write the resulting work in response.
   Answer: NOK (24, 60) = 120

The finding of the smallest common multiple (NOC) can also be done as follows. Find the LCM (12, 16, 24).

As we see from the expansion of numbers, all factors of 12 are included in the expansion of 24 (the largest of the numbers), therefore, we add only one 2 of the expansion of 16 to the LCM.

LCM (12, 16, 24) = 2 • 2 • 2 • 3 • 2 = 48

Answer: NOK (12, 16, 24) = 48

Special cases of finding the NOC

1. If one of the numbers is divided completely into the others, then the least common multiple of these numbers is equal to that number.

   For example, LCM (60, 15) = 60

2. Since mutually simple numbers do not have common prime dividers, their smallest common multiple is equal to the product of these numbers.

   Example.

   LCM (8, 9) = 72