How to find the largest multiple divisor. Finding the least common multiple: methods, examples of finding the LCM

Lancinova Aisa

Download:

Preview:

To use the preview of presentations, create an account for yourself ( account) Google and sign in: https://accounts.google.com

Slides captions:

Tasks for GCD and LCM of numbers The work of a 6th grade student of the MKOU "Kamyshovskaya OOSh" Lantsinova Aisa Supervisor Goryaeva Zoya Erdnigoryaevna, teacher of mathematics p. Kamyshovo, 2013

An example of finding the GCD of the numbers 50, 75 and 325. 1) Let's decompose the numbers 50, 75 and 325 into prime factors. 50= 2 ∙ 5 ∙ 5 75= 3 ∙ 5 ∙ 5 325= 5 ∙ 5 ∙ 13 50= 2 ∙ 5 ∙ 5 75= 3 ∙ 5 ∙ 5 325= 5 ∙ 5 ∙13 divide without a remainder the numbers a and b are called the greatest common divisor of these numbers.

An example of finding the LCM of the numbers 72, 99 and 117. 1) Let us factorize the numbers 72, 99 and 117. Write out the factors included in the expansion of one of the numbers 2 ∙ 2 ∙ 2 ∙ 3 ​​∙ 3 and add to them the missing factors of the remaining numbers. 2 ∙ 2 ∙ 2 ∙ 3 ​​∙ 3 ∙ 11 ∙ 13 3) Find the product of the resulting factors. 2 ∙ 2 ∙ 2 ∙ 3 ​​∙ 3 ∙ 11 ∙ 13= 10296 Answer: LCM (72, 99 and 117) = 10296 The least common multiple of natural numbers a and b is called the smallest natural number that is a multiple of a and b.

A sheet of cardboard has the shape of a rectangle, the length of which is 48 cm and the width is 40 cm. This sheet must be cut without waste into equal squares. What are the largest squares that can be obtained from this sheet and how many? Solution: 1) S = a ∙ b is the area of ​​the rectangle. S \u003d 48 ∙ 40 \u003d 1960 cm². is the area of ​​the cardboard. 2) a - the side of the square 48: a - the number of squares that can be laid along the length of the cardboard. 40: a - the number of squares that can be laid across the width of the cardboard. 3) GCD (40 and 48) \u003d 8 (cm) - the side of the square. 4) S \u003d a² - the area of ​​\u200b\u200bone square. S \u003d 8² \u003d 64 (cm².) - the area of ​​\u200b\u200bone square. 5) 1960: 64 = 30 (number of squares). Answer: 30 squares with a side of 8 cm each. Tasks for GCD

The fireplace in the room must be laid out with finishing tiles in the shape of a square. How many tiles will be needed for a 195 ͯ 156 cm fireplace and what are largest dimensions tiles? Solution: 1) S = 196 ͯ 156 = 30420 (cm ²) - S of the fireplace surface. 2) GCD (195 and 156) = 39 (cm) - side of the tile. 3) S = a² = 39² = 1521 (cm²) - area of ​​1 tile. 4) 30420: = 20 (pieces). Answer: 20 tiles measuring 39 ͯ 39 (cm). Tasks for GCD

A garden plot measuring 54 ͯ 48 m around the perimeter must be fenced off, for this, at regular intervals, it is necessary to put concrete pillars. How many poles must be brought for the site, and at what maximum distance from each other will the poles stand? Solution: 1) P = 2(a + b) – site perimeter. P \u003d 2 (54 + 48) \u003d 204 m. 2) GCD (54 and 48) \u003d 6 (m) - the distance between the pillars. 3) 204: 6 = 34 (pillars). Answer: 34 pillars, at a distance of 6 m. Tasks for GCD

Out of 210 burgundy, 126 white, 294 red roses, bouquets were collected, and in each bouquet the number of roses of the same color is equal. Which the largest number bouquets made from these roses and how many roses of each color are in one bouquet? Solution: 1) GCD (210, 126 and 294) = 42 (bouquets). 2) 210: 42 = 5 ( burgundy roses). 3) 126: 42 = 3 (white roses). 4) 294: 42 = 7 (red roses). Answer: 42 bouquets: 5 burgundy, 3 white, 7 red roses in each bouquet. Tasks for GCD

Tanya and Masha bought the same number mail sets. Tanya paid 90 rubles, and Masha paid 5 rubles. more. How much does one set cost? How many sets did each buy? Solution: 1) Masha paid 90 + 5 = 95 (rubles). 2) GCD (90 and 95) = 5 (rubles) - the price of 1 set. 3) 980: 5 = 18 (sets) - bought by Tanya. 4) 95: 5 = 19 (sets) - Masha bought. Answer: 5 rubles, 18 sets, 19 sets. Tasks for GCD

Three tourist boat trips start in the port city, the first of which lasts 15 days, the second - 20 and the third - 12 days. Returning to the port, the ships on the same day again go on a voyage. Motor ships left the port on all three routes today. In how many days will they sail together for the first time? How many trips will each ship make? Solution: 1) NOC (15.20 and 12) = 60 (days) - meeting time. 2) 60: 15 = 4 (voyages) - 1 ship. 3) 60: 20 = 3 (voyages) - 2 motor ship. 4) 60: 12 = 5 (voyages) - 3 motor ship. Answer: 60 days, 4 flights, 3 flights, 5 flights. Tasks for the NOC

Masha bought eggs for the Bear in the store. On the way to the forest, she realized that the number of eggs is divisible by 2,3,5,10 and 15. How many eggs did Masha buy? Solution: LCM (2;3;5;10;15) = 30 (eggs) Answer: Masha bought 30 eggs. Tasks for the NOC

It is required to make a box with a square bottom for stacking boxes measuring 16 ͯ 20 cm. What should be the shortest side of the square bottom to fit the boxes tightly into the box? Solution: 1) NOC (16 and 20) = 80 (boxes). 2) S = a ∙ b is the area of ​​1 box. S \u003d 16 ∙ 20 \u003d 320 (cm ²) - the area of ​​​​the bottom of 1 box. 3) 320 ∙ 80 = 25600 (cm ²) - square bottom area. 4) S \u003d a² \u003d a ∙ a 25600 \u003d 160 ∙ 160 - the dimensions of the box. Answer: 160 cm is the side of the square bottom. Tasks for the NOC

Along the road from point K there are power poles every 45 m. It was decided to replace these poles with others, placing them at a distance of 60 m from each other. How many poles were there and how many will they stand? Solution: 1) NOK (45 and 60) = 180. 2) 180: 45 = 4 - there were pillars. 3) 180: 60 = 3 - there were pillars. Answer: 4 pillars, 3 pillars. Tasks for the NOC

How many soldiers are marching on the parade ground if they march in formation of 12 people in a line and change into a column of 18 people in a line? Solution: 1) NOC (12 and 18) = 36 (people) - marching. Answer: 36 people. Tasks for the NOC

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.
Decision: 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.
Decision: 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.
Decision: 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.
Decision: 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) :

1. We factorize both numbers: 28 = 1 2 2 7 , 36 = 1 2 2 3 3
2. We find common factors, that is, those that both numbers have: 1, 2 and 2.
3. 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:

1. Find the product of the numbers 28 and 36: 28 36 = 1008
2. gcd(28, 36) is already known to be 4
3. 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 found for the greatest common divisor are decomposed into prime factors, then the product of the common factors is found prime factors these numbers. Also, to find the GCD of several numbers, you can use the following relationship: 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.

1. First, let's factorize the numbers: 12 = 1 2 2 3 , 32 = 1 2 2 2 2 2 , 36 = 1 2 2 3 3 .
2. Let's find common factors: 1, 2 and 2 .
3. Their product will give gcd: 1 2 2 = 4
4. Now let's find the LCM: for this we first find the LCM(12, 32): 12 32 / 4 = 96 .
5. 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 .
6. LCM(12, 32, 36) = 96 36 / 12 = 288 .

Let's start studying the least common multiple of two or more numbers. In the section, we will give a definition of the term, consider a theorem that establishes a relationship between the least common multiple and the greatest common divisor, and give examples of solving problems.

Common multiples - definition, examples

In this topic, we will be interested only in common multiples of integers other than zero.

Definition 1

Common multiple of integers is an integer that is a multiple of all given numbers. In fact, it is any integer that can be divided by any of the given numbers.

The definition of common multiples refers to two, three, or more integers.

Example 1

According to the definition given above for the number 12, the common multiples are 3 and 2. Also the number 12 will be a common multiple of the numbers 2 , 3 and 4 . The numbers 12 and -12 are common multiples of the numbers ±1, ±2, ±3, ±4, ±6, ±12.

At the same time, the common multiple for the numbers 2 and 3 will be the numbers 12 , 6 , − 24 , 72 , 468 , − 100 010 004 and a number of any others.

If we take numbers that are divisible by the first number of a pair and not divisible by the second, then such numbers will not be common multiples. So, for the numbers 2 and 3, the numbers 16 , − 27 , 5009 , 27001 will not be common multiples.

0 is a common multiple of any set of non-zero integers.

If we recall the property of divisibility with respect to opposite numbers, then it turns out that some integer k will be a common multiple of these numbers in the same way as the number - k . This means that common divisors can be either positive or negative.

Is it possible to find an LCM for all numbers?

The common multiple can be found for any integers.

Example 2

Suppose we are given k integers a 1 , a 2 , … , a k. The number that we get during the multiplication of numbers a 1 a 2 … a k according to the divisibility property, it will be divided by each of the factors that were included in the original product. This means that the product of the numbers a 1 , a 2 , … , a k is the least common multiple of these numbers.

How many common multiples can these integers have?

A group of integers can have a large number of common multiples. In fact, their number is infinite.

Example 3

Suppose we have some number k . Then the product of the numbers k · z , where z is an integer, will be a common multiple of the numbers k and z . Given that the number of numbers is infinite, then the number of common multiples is infinite.

Least Common Multiple (LCM) - Definition, Symbol and Examples

Let's remember the concept the smallest number from a given set of numbers, which we considered in the Comparison of Integers section. With this concept in mind, we formulate the definition of the least common multiple, which has the greatest practical significance among all common multiples.

Definition 2

Least common multiple of given integers is the least positive common multiple of these numbers.

The least common multiple exists for any number of given numbers. The abbreviation NOK is the most commonly used to designate a concept in the reference literature. Shorthand for Least Common Multiple for Numbers a 1 , a 2 , … , a k will look like LCM (a 1 , a 2 , … , a k).

Example 4

The least common multiple of 6 and 7 is 42. Those. LCM(6, 7) = 42. The least common multiple of four numbers - 2 , 12 , 15 and 3 will be equal to 60 . Shorthand will be LCM (- 2 , 12 , 15 , 3) ​​= 60 .

Not for all groups of given numbers, the least common multiple is obvious. Often it has to be calculated.

Relationship between NOC and NOD

The least common multiple and the greatest common divisor are related. The relationship between concepts is established by the theorem.

Theorem 1

The least common multiple of two positive integers a and b is equal to the product of the numbers a and b divided by the greatest common divisor of the numbers a and b , that is, LCM (a , b) = a b: GCD (a , b) .

Proof 1

Suppose we have some number M which is a multiple of numbers a and b . If the number M is divisible by a , there is also some integer z , under which the equality M = a k. According to the definition of divisibility, if M is also divisible by b, so then a k divided by b.

If we introduce a new notation for gcd (a , b) as d, then we can use the equalities a = a 1 d and b = b 1 · d . In this case, both equalities will be coprime numbers.

We have already established above that a k divided by b. Now this condition can be written as follows:
a 1 d k divided by b 1 d, which is equivalent to the condition a 1 k divided by b 1 according to the properties of divisibility.

According to the property mutual prime numbers, if a 1 and b 1 are mutually prime numbers, a 1 not divisible by b 1 despite the fact that a 1 k divided by b 1, then b 1 should share k.

In this case, it would be appropriate to assume that there is a number t, for which k = b 1 t, and since b1=b:d, then k = b: d t.

Now instead of k put into equality M = a k expression of the form b: d t. This allows us to come to equality M = a b: d t. At t=1 we can get the least positive common multiple of a and b , equal a b: d, provided that the numbers a and b positive.

So we have proved that LCM (a , b) = a b: GCD (a,b).

Establishing a connection between LCM and GCD allows you to find the least common multiple through the greatest common divisor of two or more given numbers.

Definition 3

The theorem has two important consequences:

• multiples of the least common multiple of two numbers are the same as common multiples of those two numbers;
• the least common multiple of coprime positive numbers a and b is equal to their product.

It is not difficult to substantiate these two facts. Any common multiple of M numbers a and b is defined by the equality M = LCM (a, b) t for some integer value t. Since a and b are coprime, then gcd (a, b) = 1, therefore, LCM (a, b) = a b: gcd (a, b) = a b: 1 = a b.

Least common multiple of three or more numbers

In order to find the least common multiple of several numbers, you must successively find the LCM of two numbers.

Theorem 2

Let's pretend that a 1 , a 2 , … , a k are some integers positive numbers. To calculate the LCM m k these numbers, we need to sequentially calculate m 2 = LCM(a 1 , a 2) , m 3 = NOC(m 2 , a 3) , … , m k = NOC(m k - 1 , a k) .

Proof 2

The first corollary of the first theorem discussed in this topic will help us to prove the correctness of the second theorem. Reasoning is built according to the following algorithm:

• common multiples of numbers a 1 and a 2 coincide with multiples of their LCM, in fact, they coincide with multiples of the number m2;
• common multiples of numbers a 1, a 2 and a 3 m2 and a 3 m 3;
• common multiples of numbers a 1 , a 2 , … , a k coincide with common multiples of numbers m k - 1 and a k, therefore, coincide with multiples of the number m k;
• due to the fact that the smallest positive multiple of the number m k is the number itself m k, then the least common multiple of the numbers a 1 , a 2 , … , a k is an m k.

So we have proved the theorem.

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

60=2*2*3*5
75=3*5*5
2) We write out the factors included in the expansion of the first of these numbers and add to them the missing factor 5 from the expansion of the second number. We get: 2*2*3*5*5=300. Found NOC, i.e. this sum = 300. Do not forget the dimension and write the answer:
Answer: Mom gives 300 rubles each.

Definition of GCD: Greatest Common Divisor (GCD) natural numbers a and in name the largest natural number c, to which and a, and b divided without remainder. Those. c is the smallest natural number for which and a and b are multiples.

Reminder: There are two approaches to the definition of natural numbers

• numbers used in: enumeration (numbering) of items (first, second, third, ...); - in schools, usually.
• indicating the number of items (no pokemon - zero, one pokemon, two pokemon, ...).

Negative and non-integer (rational, real, ...) numbers are not natural. Some authors include zero in the set of natural numbers, others do not. The set of all natural numbers is usually denoted by the symbol N

Reminder: Divisor of a natural number a call the number b, to which a divided without remainder. Multiple of natural number b called a natural number a, which is divided by b without a trace. If number b- number divisor a, then a multiple of b. Example: 2 is a divisor of 4 and 4 is a multiple of 2. 3 is a divisor of 12, and 12 is a multiple of 3.
Reminder: Natural numbers are called prime if they are divisible without remainder only by themselves and by 1. Coprime are numbers that have only one common divisor equal to 1.

Definition of how to find the GCD in the general case: To find GCD (Greatest Common Divisor) Several natural numbers are needed:
1) Decompose them into prime factors. (The Prime Number Chart can be very helpful for this.)
2) Write out the factors included in the expansion of one of them.
3) Delete those that are not included in the expansion of the remaining numbers.
4) Multiply the factors obtained in paragraph 3).

Task 2 on (NOK): By the new year, Kolya Puzatov bought 48 hamsters and 36 coffee pots in the city. Fekla Dormidontova, as the most honest girl in the class, was given the task of dividing this property into the largest possible number gift sets for teachers. What is the number of sets? What is the composition of the sets?

Example 2.1. solving the problem of finding GCD. Finding GCD by selection.
Decision: Each of the numbers 48 and 36 must be divisible by the number of gifts.
1) Write out the divisors 48: 48, 24, 16, 12 , 8, 6, 3, 2, 1
2) Write out the divisors 36: 36, 18, 12 , 9, 6, 3, 2, 1 Choose the greatest common divisor. Op-la-la! Found, this is the number of sets of 12 pieces.
3) Divide 48 by 12, we get 4, divide 36 by 12, we get 3. Do not forget the dimension and write the answer:
Answer: You will get 12 sets of 4 hamsters and 3 coffee pots in each set.

Consider three ways to find the least common multiple.

Finding by Factoring

The first way is to find the least common multiple by factoring the given numbers into prime factors.

Suppose we need to find the LCM of numbers: 99, 30 and 28. To do this, we decompose each of these numbers into prime factors:

For the desired number to be divisible by 99, 30 and 28, it is necessary and sufficient that it includes all the prime factors of these divisors. To do this, we need to take all the prime factors of these numbers to the highest occurring power and multiply them together:

2 2 3 2 5 7 11 = 13 860

So LCM (99, 30, 28) = 13,860. No other number less than 13,860 is evenly divisible by 99, 30, or 28.

To find the least common multiple of given numbers, you need to decompose them into prime factors, then take each prime factor with the largest exponent with which it occurs, and multiply these factors together.

Since coprime numbers have no common prime factors, their least common multiple is equal to the product of these numbers. For example, three numbers: 20, 49 and 33 are coprime. So

LCM (20, 49, 33) = 20 49 33 = 32,340.

The same should be done when looking for the least common multiple of various prime numbers. For example, LCM (3, 7, 11) = 3 7 11 = 231.

Finding by selection

The second way is to find the least common multiple by fitting.

Example 1. When the largest of the given numbers is evenly divisible by other given numbers, then the LCM of these numbers is equal to the larger of them. For example, given four numbers: 60, 30, 10 and 6. Each of them is divisible by 60, therefore:

NOC(60, 30, 10, 6) = 60

In other cases, to find the least common multiple is used next order actions:

1. Determine the largest number from the given numbers.
2. Next, find numbers that are multiples the largest number, multiplying it by integers in ascending order and checking whether the remaining given numbers are divisible by the resulting product.

Example 2. Given three numbers 24, 3 and 18. Determine the largest of them - this is the number 24. Next, find the numbers that are multiples of 24, checking whether each of them is divisible by 18 and by 3:

24 1 = 24 is divisible by 3 but not divisible by 18.

24 2 = 48 - divisible by 3 but not divisible by 18.

24 3 \u003d 72 - divisible by 3 and 18.

So LCM(24, 3, 18) = 72.

Finding by Sequential Finding LCM

The third way is to find the least common multiple by successively finding the LCM.

The LCM of two given numbers is equal to the product of these numbers divided by their greatest common divisor.

Example 1. Find the LCM of two given numbers: 12 and 8. Determine their greatest common divisor: GCD (12, 8) = 4. Multiply these numbers:

We divide the product into their GCD:

So LCM(12, 8) = 24.

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

1. First, the LCM of any two of the given numbers is found.
2. Then, the LCM of the found least common multiple and the third given number.
3. Then, the LCM of the resulting least common multiple and the fourth number, and so on.
4. Thus the LCM search continues as long as there are numbers.

Example 2. Let's find the LCM of three given numbers: 12, 8 and 9. We have already found the LCM of the numbers 12 and 8 in the previous example (this is the number 24). It remains to find the least common multiple of 24 and the third given number - 9. Determine their greatest common divisor: gcd (24, 9) = 3. Multiply LCM with the number 9:

We divide the product into their GCD:

So LCM(12, 8, 9) = 72.