Least common multiple (LCM). How to find the least common multiple of numbers
Let's continue the discussion about the least common multiple that we started in the LCM - Least Common Multiple, Definition, Examples section. In this topic, we will look at ways to find the LCM for three numbers or more, we will analyze the question of how to find the LCM of a negative number.
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Calculation of the least common multiple (LCM) through gcd
We have already established the relationship between the least common multiple and the greatest common divisor. Now let's learn how to define the LCM through the GCD. First, let's figure out how to do this for positive numbers.
Definition 1
Find the least common multiple through the greatest common divisor you can use the formula LCM (a, b) = a b: GCD (a, b) .
Example 1
It is necessary to find the LCM of the numbers 126 and 70.
Solution
Let's take a = 126 , b = 70 . Substitute the values in 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 the numbers 70 and 126. For this we need the Euclid algorithm: 126 = 70 1 + 56 , 70 = 56 1 + 14 , 56 = 14 4 , hence gcd (126 , 70) = 14 .
Let's calculate the LCM: LCM (126, 70) = 126 70: GCD (126, 70) = 126 70: 14 = 630.
Answer: LCM (126, 70) = 630.
Example 2
Find the nok of the numbers 68 and 34.
Solution
GCD in this case is easy to find, since 68 is divisible by 34. 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, then the LCM of these numbers will be equal to the first number.
Finding the LCM by Factoring Numbers into Prime Factors
Now let's look at a way to find the LCM, which is based on the decomposition of numbers into prime factors.
Definition 2
To find the least common multiple, we need to perform a number of simple steps:
- we make up the product of all prime factors of numbers for which we need to find the LCM;
- we exclude all prime factors from their obtained products;
- the product obtained after eliminating the common prime factors will be equal to the LCM of the given numbers.
This way 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 becomes clear: the product of the numbers a and b is equal to the product of all factors that are involved in the expansion 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 out like this: 75 = 3 5 5 and 210 = 2 3 5 7. If you make 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 , decomposing 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 the factors that participated in the expansion of these numbers will look like: 2 2 3 3 5 5 7 7 7. Let's find the common factors. This number is 7 . We exclude it from the general 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: LCM (441 , 700) = 44 100 .
Let us give one more 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 decompose 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 get the product, which will be the desired LCM of two numbers.
Example 5
Let's go back 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 number 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 decompose the numbers from the condition into prime factors: 84 = 2 2 3 7 and 648 = 2 2 2 3 3 3 3. Add to the product of 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) = 4536.
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
Suppose we have integers a 1 , a 2 , … , a k. NOC m k of these numbers is found in sequential calculation 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 specific problems.
Example 7
You need to calculate the least common multiple of the four numbers 140 , 9 , 54 and 250 .
Solution
Let's introduce the notation: a 1 \u003d 140, a 2 \u003d 9, a 3 \u003d 54, a 4 \u003d 250.
Let's start by calculating m 2 = LCM (a 1 , a 2) = LCM (140 , 9) . Let's use 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, LCM(140, 9) = 140 9: GCD(140, 9) = 140 9: 1 = 1260. Therefore, m 2 = 1 260 .
Now let's calculate according to the same algorithm m 3 = LCM (m 2 , a 3) = LCM (1 260 , 54) . In the course of calculations, we get m 3 = 3 780.
It remains for us to calculate m 4 \u003d LCM (m 3, a 4) \u003d LCM (3 780, 250) . We act according to the same algorithm. We get m 4 \u003d 94 500.
The LCM of the four numbers from the example condition is 94500 .
Answer: LCM (140, 9, 54, 250) = 94,500.
As you can see, the calculations are simple, but quite laborious. To save time, you can go the other way.
Definition 4
We offer you the following algorithm of actions:
- decompose all numbers into prime factors;
- to the product of the factors of the first number, add the missing factors from the product of the second number;
- add the missing factors of the third number to the product obtained at the previous stage, etc.;
- the resulting product will be the least common multiple of all numbers from the condition.
Example 8
It is necessary to find the LCM of five numbers 84 , 6 , 48 , 7 , 143 .
Solution
Let's decompose 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 have 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. We turn to the number 48, from the product of prime factors of which we take 2 and 2. Then we add a simple factor of 7 from the fourth number and 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 five original numbers.
Answer: LCM (84, 6, 48, 7, 143) = 48,048.
Finding the Least Common Multiple of Negative Numbers
To find the least common multiple negative numbers, these numbers must first be replaced by numbers with the opposite sign, and then the calculations should be carried out according to 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 it is accepted that a and − a- opposite numbers
then the set of multiples a coincides with 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 change 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 Euclid algorithm.
We get that the LCM of numbers − 145 and − 45 equals 1 305 .
Answer: LCM (− 145 , − 45) = 1 305 .
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But many natural numbers are evenly divisible by other natural numbers.
For example:
The number 12 is divisible by 1, by 2, by 3, by 4, by 6, by 12;
The number 36 is divisible by 1, by 2, by 3, by 4, by 6, by 12, by 18, by 36.
The numbers by which the number is divisible (for 12 it is 1, 2, 3, 4, 6 and 12) are called number divisors. Divisor of a natural number a is the natural number that divides the given number a without a trace. A natural number that has more than two factors is called composite .
Note that the numbers 12 and 36 have common divisors. These are the numbers: 1, 2, 3, 4, 6, 12. The largest divisor of these numbers is 12. The common divisor of these two numbers a and b is the number by which both given numbers are divisible without a remainder a and b.
common multiple several numbers is called the number that is divisible by each of these numbers. For example, the numbers 9, 18 and 45 have a common multiple of 180. But 90 and 360 are also their common multiples. Among all jcommon multiples, there is always the smallest one, in this case it is 90. This number is called leastcommon multiple (LCM).
LCM is always a natural number, which must be greater than the largest of the numbers for which it is defined.
Least common multiple (LCM). Properties.
Commutativity:
Associativity:
In particular, if and are coprime numbers , then:
Least common multiple of two integers m and n is a divisor of all other common multiples m and n. Moreover, the set of common multiples m,n coincides with the set of multiples for LCM( m,n).
The asymptotics for can be expressed in terms of some number-theoretic functions.
So, Chebyshev function. As well as:
This follows from the definition and properties of the Landau function g(n).
What follows from the law of distribution prime numbers.
Finding the least common multiple (LCM).
NOC( a, b) can be calculated in several ways:
1. If the greatest common divisor is known, you can use its relationship with the LCM:
2. Let the canonical decomposition of both numbers into prime factors be known:
where p 1 ,...,p k are various prime numbers, and d 1 ,...,dk and e 1 ,...,ek are non-negative integers (they can be zero if the corresponding prime is not in the decomposition).
Then LCM ( a,b) is calculated by the formula:
In other words, the LCM expansion contains all prime factors that are included in at least one of the number expansions a, b, and the largest of the two exponents of this factor is taken.
Example:
The calculation of the least common multiple of several numbers can be reduced to several successive calculations of the LCM of two numbers:
Rule. To find the LCM of a series of numbers, you need:
- decompose numbers into prime factors;
- transfer the largest expansion to the factors of the desired product (the product of the factors of the largest number of the given ones), and then add factors from the expansion of other numbers that do not occur in the first number or are in it a smaller number of times;
- the resulting product of prime factors will be the LCM of the given numbers.
Any two or more natural numbers have their own NOC. If the numbers are not multiples of each other or do not have the same factors in the expansion, then their LCM is equal to the product of these numbers.
The prime factors of the number 28 (2, 2, 7) were supplemented with a factor of 3 (the number 21), the resulting product (84) will be the smallest number that is divisible by 21 and 28.
The prime factors of the largest number 30 were supplemented with a factor of 5 of the number 25, the resulting product 150 is greater than the largest number 30 and is divisible by all given numbers without a remainder. This is the smallest possible product (150, 250, 300...) that all given numbers are multiples of.
The numbers 2,3,11,37 are prime, so their LCM is equal to the product of the given numbers.
rule. To calculate the LCM of prime numbers, you need to multiply all these numbers together.
Another option:
To find the least common multiple (LCM) of several numbers you need:
1) represent each number as a product of its prime factors, for example:
504 \u003d 2 2 2 3 3 7,
2) write down the powers of all prime factors:
504 \u003d 2 2 2 3 3 7 \u003d 2 3 3 2 7 1,
3) write down all prime divisors (multipliers) of each of these numbers;
4) choose the largest degree of each of them, found in all expansions of these numbers;
5) multiply these powers.
Example. Find the LCM of numbers: 168, 180 and 3024.
Solution. 168 \u003d 2 2 2 3 7 \u003d 2 3 3 1 7 1,
180 \u003d 2 2 3 3 5 \u003d 2 2 3 2 5 1,
3024 = 2 2 2 2 3 3 3 7 = 2 4 3 3 7 1 .
We write out the largest powers of all prime divisors and multiply them:
LCM = 2 4 3 3 5 1 7 1 = 15120.
Definition. The largest natural number by which the numbers a and b are divisible without a remainder is called greatest common divisor (gcd) these numbers.
Let's find the greatest common divisor of the numbers 24 and 35.
The divisors of 24 will be the numbers 1, 2, 3, 4, 6, 8, 12, 24, and the divisors of 35 will be the numbers 1, 5, 7, 35.
We see that the numbers 24 and 35 have only one common divisor - the number 1. Such numbers are called coprime.
Definition. The natural numbers are called coprime if their greatest common divisor (gcd) is 1.
Greatest Common Divisor (GCD) can be found without writing out all the divisors of the given numbers.
Factoring the numbers 48 and 36, we get:
48 = 2 * 2 * 2 * 2 * 3, 36 = 2 * 2 * 3 * 3.
From the factors included in the expansion of the first of these numbers, we delete those that are not included in the expansion of the second number (i.e., two deuces).
The factors 2 * 2 * 3 remain. Their product is 12. This number is the greatest common divisor of the numbers 48 and 36. The greatest common divisor of three or more numbers is also found.
To find greatest common divisor
2) from the factors included in the expansion of one of these numbers, cross out those that are not included in the expansion of other numbers;
3) find the product of the remaining factors.
If all given numbers are divisible by one of them, then this number is greatest common divisor given numbers.
For example, the greatest common divisor of 15, 45, 75, and 180 is 15, since it divides all other numbers: 45, 75, and 180.
Least common multiple (LCM)
Definition. Least common multiple (LCM) natural numbers a and b are the smallest natural number that is a multiple of both a and b. The least common multiple (LCM) of the numbers 75 and 60 can be found without writing out multiples of these numbers in a row. To do this, we decompose 75 and 60 into simple factors: 75 \u003d 3 * 5 * 5, and 60 \u003d 2 * 2 * 3 * 5.
Let's write out the factors included in the expansion of the first of these numbers, and add to them the missing factors 2 and 2 from the expansion of the second number (that is, we combine the factors).
We get five factors 2 * 2 * 3 * 5 * 5, the product of which is 300. This number is the least common multiple of the numbers 75 and 60.
Also find the least common multiple of three or more numbers.
To find the least common multiple several natural numbers, you need:
1) decompose them into prime factors;
2) write out the factors included in the expansion of one of the numbers;
3) add to them the missing factors from the expansions of the remaining numbers;
4) find the product of the resulting factors.
Note that if one of these numbers is divisible by all other numbers, then this number is the least common multiple of these numbers.
For example, the least common multiple of 12, 15, 20, and 60 would be 60, since it is divisible by all given numbers.
Pythagoras (VI century BC) and his students studied the issue of divisibility of numbers. A number equal to the sum of all its divisors (without the number itself), they called the perfect number. For example, the numbers 6 (6 = 1 + 2 + 3), 28 (28 = 1 + 2 + 4 + 7 + 14) are perfect. The next perfect numbers are 496, 8128, 33,550,336. The Pythagoreans knew only the first three perfect numbers. The fourth - 8128 - became known in the 1st century. n. e. The fifth - 33 550 336 - was found in the 15th century. By 1983, 27 perfect numbers were already known. But until now, scientists do not know whether there are odd perfect numbers, whether there is the largest perfect number.
The interest of ancient mathematicians in prime numbers is due to the fact that any number is either prime or can be represented as a product of prime numbers, that is, prime numbers are like bricks from which the rest of the natural numbers are built.
You probably noticed that prime numbers in the series of natural numbers occur unevenly - in some parts of the series there are more of them, in others - less. But the further we move along the number series, the rarer the prime numbers. The question arises: does the last (largest) prime number exist? The ancient Greek mathematician Euclid (3rd century BC), in his book "Beginnings", which for two thousand years was the main textbook of mathematics, proved that there are infinitely many prime numbers, that is, behind each prime number there is an even greater prime number.
To find prime numbers, another Greek mathematician of the same time, Eratosthenes, came up with such a method. He wrote down all the numbers from 1 to some number, and then crossed out the unit, which is neither a prime nor a composite number, then crossed out through one all the numbers after 2 (numbers that are multiples of 2, i.e. 4, 6 , 8, etc.). The first remaining number after 2 was 3. Then, after two, all the numbers after 3 were crossed out (numbers that are multiples of 3, i.e. 6, 9, 12, etc.). in the end, only the prime numbers remained uncrossed out.
The online calculator allows you to quickly find the greatest common divisor and least common multiple of two or any other number of numbers.
Calculator for finding GCD and NOC
Find GCD and NOC
GCD and NOC found: 5806
How to use the calculator
- Enter numbers in the input field
- In case of entering incorrect characters, the input field will be highlighted in red
- press the button "Find GCD and NOC"
How to enter numbers
- Numbers are entered separated by spaces, dots or commas
- The length of the entered numbers is not limited, so finding the gcd and lcm of long numbers will not be difficult
What is NOD and NOK?
Greatest Common Divisor of several numbers is the largest natural integer by which all the original numbers are divisible without a remainder. The greatest common divisor is abbreviated as GCD.
Least common multiple several numbers is the smallest number that is divisible by each of the original numbers without a remainder. The least common multiple is abbreviated as NOC.
How to check if a number is divisible by another number without a remainder?
To find out if one number is divisible by another without a remainder, you can use some properties of divisibility of numbers. Then, by combining them, one can check the divisibility by some of them and their combinations.
Some signs of divisibility of numbers
1. Sign of divisibility of a number by 2
To determine whether a number is divisible by two (whether it is even), it is enough to look at the last digit of this number: if it is equal to 0, 2, 4, 6 or 8, then the number is even, which means it is divisible by 2.
Example: determine if the number 34938 is divisible by 2.
Solution: look at the last digit: 8 means the number is divisible by two.
2. Sign of divisibility of a number by 3
A number is divisible by 3 when the sum of its digits is divisible by 3. Thus, to determine if a number is divisible by 3, you need to calculate the sum of the digits and check if it is divisible by 3. Even if the sum of the digits turned out to be very large, you can repeat the same process again.
Example: determine if the number 34938 is divisible by 3.
Solution: we count the sum of the digits: 3+4+9+3+8 = 27. 27 is divisible by 3, which means that the number is divisible by three.
3. Sign of divisibility of a number by 5
A number is divisible by 5 when its last digit is zero or five.
Example: determine if the number 34938 is divisible by 5.
Solution: look at the last digit: 8 means the number is NOT divisible by five.
4. Sign of divisibility of a number by 9
This sign is very similar to the sign of divisibility by three: a number is divisible by 9 when the sum of its digits is divisible by 9.
Example: determine if the number 34938 is divisible by 9.
Solution: we calculate the sum of the digits: 3+4+9+3+8 = 27. 27 is divisible by 9, which means that the number is divisible by nine.
How to find GCD and LCM of two numbers
How to find the GCD of two numbers
Most in a simple way calculating the greatest common divisor of two numbers is to find all possible divisors of those numbers and choose the largest of them.
Consider this method using the example of finding GCD(28, 36) :
- We factorize both numbers: 28 = 1 2 2 7 , 36 = 1 2 2 3 3
- We find common factors, that is, those that both numbers have: 1, 2 and 2.
- We calculate the product of these factors: 1 2 2 \u003d 4 - this is the greatest common divisor of the numbers 28 and 36.
How to find the LCM of two numbers
There are two most common ways to find the smallest multiple of two numbers. The first way is that you can write out the first multiples of two numbers, and then choose among them such a number that will be common to both numbers and at the same time the smallest. And the second is to find the GCD of these numbers. Let's just consider it.
To calculate the LCM, you need to calculate the product of the original numbers and then divide it by the previously found GCD. Let's find the LCM for the same numbers 28 and 36:
- Find the product of the numbers 28 and 36: 28 36 = 1008
- gcd(28, 36) is already known to be 4
- LCM(28, 36) = 1008 / 4 = 252 .
Finding GCD and LCM for Multiple Numbers
The greatest common divisor can be found for several numbers, and not just for two. For this, the numbers to be searched for the greatest common divisor are decomposed into prime factors, then the product of the common prime factors of these numbers is found. Also, to find the GCD of several numbers, you can use the following relation: gcd(a, b, c) = gcd(gcd(a, b), c).
A similar relation also applies to the least common multiple of numbers: LCM(a, b, c) = LCM(LCM(a, b), c)
Example: find GCD and LCM for numbers 12, 32 and 36.
- First, let's factorize the numbers: 12 = 1 2 2 3 , 32 = 1 2 2 2 2 2 , 36 = 1 2 2 3 3 .
- Let's find common factors: 1, 2 and 2 .
- Their product will give gcd: 1 2 2 = 4
- Now let's find the LCM: for this we first find the LCM(12, 32): 12 32 / 4 = 96 .
- To find the NOC of all three numbers, you need to find gcd(96, 36): 96 = 1 2 2 2 2 2 3 , 36 = 1 2 2 3 3 , gcd = 1 2 2 3 = 12 .
- LCM(12, 32, 36) = 96 36 / 12 = 288 .
Mathematical expressions and tasks require a lot of additional knowledge. NOC is one of the main ones, especially often used in the topic. The topic is studied in high school, while it is not particularly difficult to understand material, it will not be difficult for a person familiar with powers and the multiplication table to select the necessary numbers and find the result.
Definition
A common multiple is 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.
NOC is the accepted term for short title, 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 into factors, the larger the number, the more factors there 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 option is much more difficult. The numbers 300 and 1260 are given, finding the LCM is mandatory. To solve the task, the following actions are assumed:
Decomposition of the first and second numbers into the simplest factors. 300 = 2 2 * 3 * 5 2 ; 1260 = 2 2 * 3 2 * 5 * 7. The first stage has been completed.
The second stage involves working with the already obtained data. 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 instance. Both initial numbers have in their composition the numbers 2, 3 and 5, in different degrees, 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 correct filling The task fits into two steps without explanation:
1) 300 = 2 2 * 3 * 5 2 ; 1260 = 2 2 * 3 2 *5 *7.
2) NOK = 6300.
That's the whole problem, if you try to calculate the right number through multiplication, then the answer will definitely not be correct, since 300 * 1260 = 378,000.
Examination:
6300 / 300 = 21 - true;
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 NOC mean in mathematics
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 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. How 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 LCM:
1) 250 = 25 * 10 = 5 2 * 5 * 2 = 5 3 * 2 - this example describes the factorization in detail, 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 required to mention all the factors, in this case 2, 5, 3 are given - for all these numbers it is required to determine the maximum degree.
Attention: all multipliers must be brought to full simplification, if possible, decomposing to the level of single digits.
Examination:
1) 3000 / 250 = 12 - true;
2) 3000 / 600 = 5 - true;
3) 3000 / 1500 = 2 is correct.
This method does not require any tricks or genius level abilities, everything is simple and clear.
Another way
In mathematics, a lot is connected, a lot 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.
Given the numbers 30, 35, 42, you need to find the LCM that connects 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 the processes associated with this calculation, there is also the greatest common divisor, which is calculated according to similar principles and is often encountered in neighboring problems. The difference is small, but significant enough, the LCM involves the calculation of a number that is divisible by all given initial values, and the GCM involves the calculation greatest value by which the original numbers are divisible.
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