Nok definition and properties. Least Common Multiple (LCM) – Definition, Examples and Properties. Finding NOC via GCD

A multiple is a number that is divisible by a given number without a remainder. The least common multiple (LCM) of a group of numbers is the smallest number that is divisible by each number in the group without leaving a remainder. To find the least common multiple, you need to find the prime factors of given numbers. The LCM can also be calculated using a number of other methods that apply to groups of two or more numbers.

Steps

Series of multiples

Look at these numbers. The method described here is best used when given two numbers, each of which is less than 10. If larger numbers are given, use a different method.

For example, find the least common multiple of 5 and 8. These are small numbers, so you can use this method.

A multiple is a number that is divisible by a given number without a remainder. Multiples can be found in the multiplication table.
- For example, numbers that are multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40.
Write down a series of numbers that are multiples of the first number. Do this under multiples of the first number to compare two sets of numbers.
- For example, numbers that are multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, and 64.
Find the smallest number that is present in both sets of multiples. You may have to write long series of multiples to find the total number. The smallest number that is present in both sets of multiples is the least common multiple.
- For example, the smallest number that appears in the series of multiples of 5 and 8 is the number 40. Therefore, 40 is the least common multiple of 5 and 8.
Prime factorization
1. Look at these numbers. The method described here is best used when given two numbers, each of which is greater than 10. If smaller numbers are given, use a different method.
  - For example, find the least common multiple of the numbers 20 and 84. Each of the numbers is greater than 10, so you can use this method.
2. Factor into prime factors first number. That is, you need to find such prime numbers that, when multiplied, will result in a given number. Once you have found the prime factors, write them as equalities.
  
  Factor the second number into prime factors. Do this in the same way as you factored the first number, that is, find such prime numbers that, when multiplied, will yield the given number.
  
  Write down the factors common to both numbers. Write such factors as a multiplication operation. As you write each factor, cross it out in both expressions (expressions that describe factorizations of numbers into prime factors).
  
  Add the remaining factors to the multiplication operation. These are factors that are not crossed out in both expressions, that is, factors that are not common to both numbers.
  
  Calculate the least common multiple. To do this, multiply the numbers in the written multiplication operation.
Finding common factors
1. Find the divisor common to both numbers. Write it down in the first row and first column. It is better to look for prime factors, but this is not a requirement.
  - For example, 18 and 30 are even numbers, so their common factor is 2. So write 2 in the first row and first column.
2. Divide each number by the first divisor. Write each quotient under the appropriate number. A quotient is the result of dividing two numbers.
  
  Find the divisor common to both quotients. If there is no such divisor, skip the next two steps. Otherwise, write the divisor in the second row and first column.
  - For example, 9 and 15 are divisible by 3, so write 3 in the second row and first column.
3. Divide each quotient by its second divisor. Write each division result under the corresponding quotient.
  
  If necessary, add additional cells to the grid. Repeat the described steps until the quotients have a common divisor.
  
  Circle the numbers in the first column and last row of the grid. Then write the selected numbers as a multiplication operation.
Euclid's algorithm

A number can be a multiple of not one, but several numbers at once, such a number is called common multiple given numbers.

Example. The numbers 3 are multiples of: 6, 9, 12 , 15, etc. The number 4 is a multiple of the number: 8, 12 , 16, 20, etc. You can notice that the same number (12) is divisible by both numbers 3 and 4. Therefore, the number 12 is a common multiple of the numbers 3 and 4.

Common multiple numbers is any number that is divisible without a remainder by each of the given numbers.

Finding the common multiple of several natural numbers is quite easy; you can simply multiply the given numbers, the resulting product will be their common multiple.

Example. Find the common multiple of the numbers 2, 3, 4, 6.

Solution:

2 3 4 6 = 144

The number 144 is a common multiple of the numbers 2, 3, 4 and 6.

For any number of natural numbers, there are infinitely many multiples.

Example. For the numbers 12 and 20, the multiples are: 60, 120, 180, 240, etc. These are all common multiples of the numbers 12 and 20.

Least common multiple

Least common multiple (LCM) several numbers - this is the smallest natural number that is divisible without a remainder by each of these numbers.

Example. The least common multiple of 3, 4 and 9 is 36; no other number less than 36 is divisible by 3, 4 and 9 without a remainder.

The least common multiple is written as follows: LCM ( a, b, ...). The numbers in parentheses can be listed in any order.

Example. Let's write down the least common multiple of the numbers 3, 4 and 9:

LCM(3, 4, 9) = 36

How to find NOC

Let's consider two ways to find the least common multiple: using the decomposition of numbers into prime factors and finding the LCM through the GCD.

Using prime factorization

To find the LCM of several natural numbers, you need to decompose these numbers into prime factors, then take from these decompositions each prime factor with the largest exponent and multiply these factors among themselves.

Example.

Solution:

99 = 3 3 11 = 3 2 11

54 = 2 3 3 3 = 2 3 3

The least common multiple must be divisible by 99, which means that it must include all the factors of the number 99. Further, the LCM must also be divisible by 54, i.e., it must also include the factors of this number.

Let us write out from these expansions each prime factor with the largest exponent and multiply these factors among themselves. We get the following product:

2 3 3 11 = 594

This is the least common multiple of these numbers. No other number less than 594 is divisible by 99 and 54.

Answer: LCM(99, 54) = 594.

Since coprime numbers do not have identical prime factors, their least common multiple is equal to the product of these numbers.

Example. Find the least common multiple of two numbers 12 and 49.

Solution:

Let's factor each of these numbers into prime factors:

12 = 2 2 3 = 2 2 3
49 = 7 7 = 7 2

Applying the rule to this case, we come to the conclusion that coprime numbers must simply be multiplied:

2 2 3 7 2 = 12 49 = 980

Answer: LCM(12, 49) = 980.

You should do the same thing when you need to find the least common multiple of prime numbers.

Example. Find the least common multiple of 5, 7 and 13.

Solution:

Since these numbers are prime, we simply multiply them:

5 7 13 = 455

Answer: LCM(5, 7, 13) = 455.

If the largest of the given numbers is divisible by all other numbers, then this number will be the least common multiple of the given numbers.

Example. Find the least common multiple of 24, 12 and 4.

Solution:

Let's factor each of these numbers into prime factors:

24 = 2 2 2 3 = 2 3 3
12 = 2 2 3 = 2 2 3
4 = 2 2 = 2 2

You can notice that the decomposition of the larger number contains all the factors of the remaining numbers, which means that the largest of these numbers is divisible by all other numbers (including itself) and is the least common multiple:

Answer: LCM(24, 12, 4) = 24.

Finding NOC via GCD

The LCM of two natural numbers is equal to the product of these numbers divided by their GCD.

The general rule is:

NOC ( m, n) = m · n: GCD ( m, n)

Example. Find the least common multiple of two numbers 99 and 54.

Solution:

First we find their greatest common divisor:

GCD (99, 54) = 9.

Now we can calculate the LCM of these numbers using the formula:

LCM(99, 54) = 99 54: GCD(99, 54) = 5346: 9 = 594

Answer: LCM(99, 54) = 594.

To find the LCM of three or more numbers, use the following procedure:

Find the LCM of any two of the given numbers.
Then find the least common multiple of the found LCM and the third number, etc.
Thus, the search for LCM continues as long as there are numbers.

Example. Find the least common multiple of 8, 12 and 9.

Solution:

First we find the greatest common divisor of any two of these numbers, for example, 12 and 8:

GCD (12, 8) = 4.

We calculate their LCM using the formula:

LCD (12, 8) = 12 8: GCD (12, 8) = 96: 4 = 24

Now let’s find the LCM of the number 24 and the remaining number 9. Their GCM:

GCD (24, 9) = 3.

We calculate the LOC using the formula:

LCD (24, 9) = 24 9: GCD (24, 9) = 216: 3 = 72

Answer: LCM(8, 12, 9) = 72.

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Let's continue the conversation about the least common multiple, which we started in the section “LCM - least common multiple, definition, examples.” In this topic, we will look at ways to find the LCM for three or more numbers, and we will look at the question of how to find the LCM of a negative number.

Calculating Least Common Multiple (LCM) via GCD

We have already established the relationship between the least common multiple and the greatest common divisor. Now let's learn how to determine the LCM through GCD. First, let's figure out how to do this for positive numbers.

Definition 1

You can find the least common multiple through the greatest common divisor using the formula LCM (a, b) = a · b: GCD (a, b).

Example 1

You need to find the LCM of the numbers 126 and 70.

Solution

Let's take a = 126, b = 70. Let's substitute the values into the formula for calculating the least common multiple through the greatest common divisor LCM (a, b) = a · b: GCD (a, b) .

Finds the gcd of numbers 70 and 126. For this we need the Euclidean algorithm: 126 = 70 1 + 56, 70 = 56 1 + 14, 56 = 14 4, therefore GCD (126 , 70) = 14 .

Let's calculate the LCM: LCD (126, 70) = 126 70: GCD (126, 70) = 126 70: 14 = 630.

Answer: LCM(126, 70) = 630.

Example 2

Find the number 68 and 34.

Solution

GCD in this case is not difficult to find, since 68 is divisible by 34. Let's calculate the least common multiple using the formula: LCM (68, 34) = 68 34: GCD (68, 34) = 68 34: 34 = 68.

Answer: LCM(68, 34) = 68.

In this example, we used the rule for finding the least common multiple of positive integers a and b: if the first number is divisible by the second, the LCM of those numbers will be equal to the first number.

Finding the LCM by factoring numbers into prime factors

Now let's look at the method of finding the LCM, which is based on factoring numbers into prime factors.

Definition 2

To find the least common multiple, we need to perform a number of simple steps:

we compose the product of all prime factors of the numbers for which we need to find the LCM;
we exclude all prime factors from their resulting products;
the product obtained after eliminating the common prime factors will be equal to the LCM of the given numbers.

This method of finding the least common multiple is based on the equality LCM (a, b) = a · b: GCD (a, b). If you look at the formula, it will become clear: the product of the numbers a and b is equal to the product of all the factors that participate in the decomposition of these two numbers. In this case, the gcd of two numbers is equal to the product of all prime factors that are simultaneously present in the factorizations of these two numbers.

Example 3

We have two numbers 75 and 210. We can factor them as follows: 75 = 3 5 5 And 210 = 2 3 5 7. If you compose the product of all the factors of the two original numbers, you get: 2 3 3 5 5 5 7.

If we exclude the factors common to both numbers 3 and 5, we get a product of the following form: 2 3 5 5 7 = 1050. This product will be our LCM for the numbers 75 and 210.

Example 4

Find the LCM of numbers 441 And 700 , factoring both numbers into prime factors.

Solution

Let's find all the prime factors of the numbers given in the condition:

441 147 49 7 1 3 3 7 7

700 350 175 35 7 1 2 2 5 5 7

We get two chains of numbers: 441 = 3 3 7 7 and 700 = 2 2 5 5 7.

The product of all factors that participated in the decomposition of these numbers will have the form: 2 2 3 3 5 5 7 7 7. Let's find common factors. This is the number 7. Let's exclude it from the total product: 2 2 3 3 5 5 7 7. It turns out that NOC (441, 700) = 2 2 3 3 5 5 7 7 = 44 100.

Answer: LOC(441, 700) = 44,100.

Let us give another formulation of the method for finding the LCM by decomposing numbers into prime factors.

Definition 3

Previously, we excluded from the total number of factors common to both numbers. Now we will do it differently:

Let's factor both numbers into prime factors:
add to the product of the prime factors of the first number the missing factors of the second number;
we obtain the product, which will be the desired LCM of two numbers.

Example 5

Let's return to the numbers 75 and 210, for which we already looked for the LCM in one of the previous examples. Let's break them down into simple factors: 75 = 3 5 5 And 210 = 2 3 5 7. To the product of factors 3, 5 and 5 numbers 75 add the missing factors 2 And 7 numbers 210. We get: 2 · 3 · 5 · 5 · 7 . This is the LCM of the numbers 75 and 210.

Example 6

It is necessary to calculate the LCM of the numbers 84 and 648.

Solution

Let's factor the numbers from the condition into simple factors: 84 = 2 2 3 7 And 648 = 2 2 2 3 3 3 3. Let's add to the product the factors 2, 2, 3 and 7 numbers 84 missing factors 2, 3, 3 and
3 numbers 648. We get the product 2 2 2 3 3 3 3 7 = 4536. This is the least common multiple of 84 and 648.

Answer: LCM(84, 648) = 4,536.

Finding the LCM of three or more numbers

Regardless of how many numbers we are dealing with, the algorithm of our actions will always be the same: we will sequentially find the LCM of two numbers. There is a theorem for this case.

Theorem 1

Let's assume we have integers a 1 , a 2 , … , a k. NOC m k these numbers are found by sequentially calculating m 2 = LCM (a 1, a 2), m 3 = LCM (m 2, a 3), ..., m k = LCM (m k − 1, a k).

Now let's look at how the theorem can be applied to solve specific problems.

Example 7

You need to calculate the least common multiple of four numbers 140, 9, 54 and 250 .

Solution

Let us introduce the notation: a 1 = 140, a 2 = 9, a 3 = 54, a 4 = 250.

Let's start by calculating m 2 = LCM (a 1 , a 2) = LCM (140, 9). Let's apply the Euclidean algorithm to calculate the GCD of the numbers 140 and 9: 140 = 9 15 + 5, 9 = 5 1 + 4, 5 = 4 1 + 1, 4 = 1 4. We get: GCD (140, 9) = 1, GCD (140, 9) = 140 9: GCD (140, 9) = 140 9: 1 = 1,260. Therefore, m 2 = 1,260.

Now let’s calculate using the same algorithm m 3 = LCM (m 2 , a 3) = LCM (1 260, 54). During the calculations we obtain m 3 = 3 780.

We just have to calculate m 4 = LCM (m 3 , a 4) = LCM (3 780, 250). We follow the same algorithm. We get m 4 = 94 500.

The LCM of the four numbers from the example condition is 94500.

Answer: NOC (140, 9, 54, 250) = 94,500.

As you can see, the calculations are simple, but quite labor-intensive. To save time, you can go another way.

Definition 4

We offer you the following algorithm of actions:

we decompose all numbers into prime factors;
to the product of the factors of the first number we add the missing factors from the product of the second number;
to the product obtained at the previous stage we add the missing factors of the third number, etc.;
the resulting product will be the least common multiple of all numbers from the condition.

Example 8

You need to find the LCM of five numbers 84, 6, 48, 7, 143.

Solution

Let's factor all five numbers into prime factors: 84 = 2 2 3 7, 6 = 2 3, 48 = 2 2 2 2 3, 7, 143 = 11 13. Prime numbers, which is the number 7, cannot be factored into prime factors. Such numbers coincide with their decomposition into prime factors.

Now let's take the product of the prime factors 2, 2, 3 and 7 of the number 84 and add to them the missing factors of the second number. We decomposed the number 6 into 2 and 3. These factors are already in the product of the first number. Therefore, we omit them.

We continue to add the missing multipliers. Let's move on to the number 48, from the product of whose prime factors we take 2 and 2. Then we add the prime factor of 7 from the fourth number and the factors of 11 and 13 of the fifth. We get: 2 2 2 2 3 7 11 13 = 48,048. This is the least common multiple of the original five numbers.

Answer: LCM(84, 6, 48, 7, 143) = 48,048.

Finding the least common multiple of negative numbers

In order to find the least common multiple of negative numbers, these numbers must first be replaced by numbers with the opposite sign, and then the calculations must be carried out using the above algorithms.

Example 9

LCM (54, − 34) = LCM (54, 34) and LCM (− 622, − 46, − 54, − 888) = LCM (622, 46, 54, 888).

Such actions are permissible due to the fact that if we accept that a And − a– opposite numbers,
then the set of multiples of a number a matches the set of multiples of a number − a.

Example 10

It is necessary to calculate the LCM of negative numbers − 145 And − 45 .

Solution

Let's replace the numbers − 145 And − 45 to their opposite numbers 145 And 45 . Now, using the algorithm, we calculate the LCM (145, 45) = 145 · 45: GCD (145, 45) = 145 · 45: 5 = 1,305, having previously determined the GCD using the Euclidean algorithm.

We get that the LCM of the numbers is − 145 and − 45 equals 1 305 .

Answer: LCM (− 145, − 45) = 1,305.

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LCM - least common multiple. A number that will divide all given numbers without a remainder.

For example, if the given numbers are 2, 3, 5, then LCM=2*3*5=30

And if the given numbers are 2,4,8, then LCM =8

what is GCD?

GCD is the greatest common divisor. A number that can be used to divide each of the given numbers without leaving a remainder.

It is logical that if the given numbers are prime, then the gcd is equal to one.

And if the given numbers are 2, 4, 8, then GCD is equal to 2.

We will not describe it in general terms, but will simply show the solution with an example.

Given two numbers 126 and 44. Find GCD.

Then if we are given two numbers of the form

Then GCD is calculated as

where min is the minimum value of all powers of the number pn

and NOC as

where max is the maximum value of all powers of the number pn

Looking at the above formulas, you can easily prove that the gcd of two or more numbers will be equal to one, when among at least one pair of given values there are relatively prime numbers.

Therefore, it is easy to answer the question of what the gcd of such numbers as 3, 25412, 3251, 7841, 25654, 7 is equal to without calculating anything.

numbers 3 and 7 are coprime, and therefore gcd = 1

Let's look at an example.

Given three numbers 24654, 25473 and 954

Each number is decomposed into the following factors

Or, if we write it in an alternative form

That is, the gcd of these three numbers is equal to three

Well, we can calculate the LCM in a similar way, and it is equal to

Our bot will help you calculate the GCD and LCM of any integers, two, three or ten.

Let's find the greatest common divisor of GCD (36; 24)

Solution steps

Method No. 1

36 - composite number
24 - composite number

Let's expand the number 36

36: 2 = 18
18: 2 = 9 - divisible by the prime number 2
9: 3 = 3 - divisible by the prime number 3.

Let's break down the number 24 into prime factors and highlight them in green. We begin to select a divisor from prime numbers, starting with the smallest prime number 2, until the quotient turns out to be a prime number

24: 2 = 12 - divisible by the prime number 2
12: 2 = 6 - divisible by the prime number 2
6: 2 = 3
We complete the division since 3 is a prime number

2) Highlight it in blue and write out the common factors

36 = 2 ⋅ 2 ⋅ 3 ⋅ 3
24 = 2 ⋅ 2 ⋅ 2 ⋅ 3
Common factors (36; 24): 2, 2, 3

3) Now, to find the GCD you need to multiply the common factors

Answer: GCD (36; 24) = 2 ∙ 2 ∙ 3 = 12

Method No. 2

1) Find all possible divisors of the numbers (36; 24). To do this, we will alternately divide the number 36 into divisors from 1 to 36, and the number 24 into divisors from 1 to 24. If the number is divisible without a remainder, then we write the divisor in the list of divisors.

For number 36
36: 1 = 36; 36: 2 = 18; 36: 3 = 12; 36: 4 = 9; 36: 6 = 6; 36: 9 = 4; 36: 12 = 3; 36: 18 = 2; 36: 36 = 1;

For the number 24 Let's write down all the cases when it is divisible without a remainder:
24: 1 = 24; 24: 2 = 12; 24: 3 = 8; 24: 4 = 6; 24: 6 = 4; 24: 8 = 3; 24: 12 = 2; 24: 24 = 1;

2) Let’s write down all the common divisors of the numbers (36; 24) and highlight the largest one in green, this will be the greatest common divisor of the gcd of the numbers (36; 24)

Common factors of numbers (36; 24): 1, 2, 3, 4, 6, 12

Answer: GCD (36 ; 24) = 12

Let's find the least common multiple of the LCM (52; 49)

Solution steps

Method No. 1

1) Let's factor the numbers into prime factors. To do this, let’s check whether each of the numbers is prime (if a number is prime, then it cannot be decomposed into prime factors, and it is itself a decomposition)

52 - composite number
49 - composite number

Let's expand the number 52 into prime factors and highlight them in green. We begin to select a divisor from prime numbers, starting with the smallest prime number 2, until the quotient turns out to be a prime number

52: 2 = 26 - divisible by the prime number 2
26: 2 = 13 - divisible by the prime number 2.
We complete the division since 13 is a prime number

Let's expand the number 49 into prime factors and highlight them in green. We begin to select a divisor from prime numbers, starting with the smallest prime number 2, until the quotient turns out to be a prime number

49: 7 = 7 - divisible by the prime number 7.
We complete the division since 7 is a prime number

2) First of all, write down the factors of the largest number, and then the smaller number. Let's find the missing factors, highlight in blue in the expansion of the smaller number the factors that were not included in the expansion of the larger number.

52 = 2 ∙ 2 ∙ 13
49 = 7 ∙ 7

3) Now, to find the LCM you need to multiply the factors of the larger number with the missing factors, which are highlighted in blue

LCM (52 ; 49) = 2 ∙ 2 ∙ 13 ∙ 7 ∙ 7 = 2548

Method No. 2

1) Find all possible multiples of the numbers (52; 49). To do this, we will alternately multiply the number 52 by the numbers from 1 to 49, and the number 49 by the numbers from 1 to 52.

Select all multiples 52 in green:

52 ∙ 1 = 52 ; 52 ∙ 2 = 104 ; 52 ∙ 3 = 156 ; 52 ∙ 4 = 208 ;
52 ∙ 5 = 260 ; 52 ∙ 6 = 312 ; 52 ∙ 7 = 364 ; 52 ∙ 8 = 416 ;
52 ∙ 9 = 468 ; 52 ∙ 10 = 520 ; 52 ∙ 11 = 572 ; 52 ∙ 12 = 624 ;
52 ∙ 13 = 676 ; 52 ∙ 14 = 728 ; 52 ∙ 15 = 780 ; 52 ∙ 16 = 832 ;
52 ∙ 17 = 884 ; 52 ∙ 18 = 936 ; 52 ∙ 19 = 988 ; 52 ∙ 20 = 1040 ;
52 ∙ 21 = 1092 ; 52 ∙ 22 = 1144 ; 52 ∙ 23 = 1196 ; 52 ∙ 24 = 1248 ;
52 ∙ 25 = 1300 ; 52 ∙ 26 = 1352 ; 52 ∙ 27 = 1404 ; 52 ∙ 28 = 1456 ;
52 ∙ 29 = 1508 ; 52 ∙ 30 = 1560 ; 52 ∙ 31 = 1612 ; 52 ∙ 32 = 1664 ;
52 ∙ 33 = 1716 ; 52 ∙ 34 = 1768 ; 52 ∙ 35 = 1820 ; 52 ∙ 36 = 1872 ;
52 ∙ 37 = 1924 ; 52 ∙ 38 = 1976 ; 52 ∙ 39 = 2028 ; 52 ∙ 40 = 2080 ;
52 ∙ 41 = 2132 ; 52 ∙ 42 = 2184 ; 52 ∙ 43 = 2236 ; 52 ∙ 44 = 2288 ;
52 ∙ 45 = 2340 ; 52 ∙ 46 = 2392 ; 52 ∙ 47 = 2444 ; 52 ∙ 48 = 2496 ;
52 ∙ 49 = 2548 ;

Select all multiples 49 in green:

49 ∙ 1 = 49 ; 49 ∙ 2 = 98 ; 49 ∙ 3 = 147 ; 49 ∙ 4 = 196 ;
49 ∙ 5 = 245 ; 49 ∙ 6 = 294 ; 49 ∙ 7 = 343 ; 49 ∙ 8 = 392 ;
49 ∙ 9 = 441 ; 49 ∙ 10 = 490 ; 49 ∙ 11 = 539 ; 49 ∙ 12 = 588 ;
49 ∙ 13 = 637 ; 49 ∙ 14 = 686 ; 49 ∙ 15 = 735 ; 49 ∙ 16 = 784 ;
49 ∙ 17 = 833 ; 49 ∙ 18 = 882 ; 49 ∙ 19 = 931 ; 49 ∙ 20 = 980 ;
49 ∙ 21 = 1029 ; 49 ∙ 22 = 1078 ; 49 ∙ 23 = 1127 ; 49 ∙ 24 = 1176 ;
49 ∙ 25 = 1225 ; 49 ∙ 26 = 1274 ; 49 ∙ 27 = 1323 ; 49 ∙ 28 = 1372 ;
49 ∙ 29 = 1421 ; 49 ∙ 30 = 1470 ; 49 ∙ 31 = 1519 ; 49 ∙ 32 = 1568 ;
49 ∙ 33 = 1617 ; 49 ∙ 34 = 1666 ; 49 ∙ 35 = 1715 ; 49 ∙ 36 = 1764 ;
49 ∙ 37 = 1813 ; 49 ∙ 38 = 1862 ; 49 ∙ 39 = 1911 ; 49 ∙ 40 = 1960 ;
49 ∙ 41 = 2009 ; 49 ∙ 42 = 2058 ; 49 ∙ 43 = 2107 ; 49 ∙ 44 = 2156 ;
49 ∙ 45 = 2205 ; 49 ∙ 46 = 2254 ; 49 ∙ 47 = 2303 ; 49 ∙ 48 = 2352 ;
49 ∙ 49 = 2401 ; 49 ∙ 50 = 2450 ; 49 ∙ 51 = 2499 ; 49 ∙ 52 = 2548 ;

2) Let’s write down all the common multiples of the numbers (52; 49) and highlight the smallest one in green, this will be the smallest common multiple of the numbers (52; 49).

Common multiples of numbers (52; 49): 2548

Answer: LCM (52; 49) = 2548