What is the node of coprime numbers. Greatest Common Divisor

The natural numbers a and b are called coprime, if their greatest common divisor is 1 (gcd(a ; b ) = 1). In other words, if the numbers a and b have no common divisors other than 1, then they are coprime.

Examples of pairs of coprime numbers: 2 and 5, 13 and 16, 35 and 88, etc. You can specify several coprime numbers, for example, the numbers 7, 9, 16 are coprime.

Often coprime numbers are denoted as follows: (a, b) \u003d 1. For example, (23, 30) \u003d 1. This entry is, as it were, an abbreviation for the designation of the greatest common divisor of two numbers (GCD (23, 30) \u003d 1), and says that their greatest common divisor is 1.

Two adjacent natural numbers will always be coprime. For example, 15 and 16 are a pair of coprime numbers, just like 16 and 17. This is easy to understand if we take into account the "rule" that if two natural numbers a and b are divisible by the same natural number greater than 1 ( n > 1), then their difference must also be divisible by this number n (here we mean that a, b and their difference are divided by an integer, that is, they are multiples of the number n). But if a and b are two adjacent numbers (let a< b ), то b – a = 1; но 1 делится только на 1 (из ряда натуральных чисел). Следовательно, a и b не имеют других общих делителей, кроме 1.

It also follows from the definition of coprime numbers and prime numbers that different prime numbers are always coprime. After all, the only divisors of any prime number are itself and 1.

Properties of coprime numbers

• The least common multiple (LCM) of a pair of coprime numbers is equal to their product. For example, (3, 8) = 1 (which means relatively prime), so their LCM is 3 × 8 = 24 (LCM(3, 8) = 24). Indeed, you will not find a smaller number than 24 that is a multiple of both 3 and 8.
• If the numbers a and b are coprime and the number c is a multiple of both a and b, then this number will also be a multiple of the product of ab. This can be written as follows: if c a and c b , then c ab . For example, (3, 10) = 1, the number 60 is a multiple of both 3 and 10, and is also a multiple of 30 (3 × 10).
• If the numbers a and b are coprime and the number c is taken as a multiple of b (c b ), then the product ac will also be a multiple of b (ac b ). For example, (2, 17) = 1, let c = 34. The number 34 is a multiple of b = 17, then ac = 2 × 34 = 68. Check: 68 ÷ 17 = 4, i.e., it is divisible, which means 68 is a multiple 17.

Usually there are more properties than listed here. In addition, the properties of relatively prime numbers are formulated in different ways. It is also sometimes required to prove these properties (in this case no proofs are given).

The greatest common divisor of coprime numbers is always one.

Examples of nodes of relatively prime numbers.

GCD of numbers 11 and 7

The numbers 11 and 7 are coprime and, at the same time, prime.

The numbers 11 and 7 have no other common divisors other than 1.

gcd(11, 7) = 1

GCD of numbers 11 and 15

The numbers 11 and 15 are relatively prime. In this case, 11 is a prime number, and 15 is a composite number.

The divisors of 11 are 1 and 11.

The divisors of 15 are 1, 3, 5, 15.

As you can see, the only common factor of the numbers 11 and 15 is the number 1. The unit, therefore, is the GCD of the numbers 11 and 15:

gcd(11, 15) = 1

GCD of numbers 10 and 21

The numbers 10 and 21 are relatively prime. In this case, both the number 10 and the number 21 are composite.

The factors of 10 are 1, 2, 5, 10.

The factors of the number 21 are 1, 3, 7, 21.

As you can see, the only common factor of the numbers 10 and 21 is the number 1. The unit, therefore, is the GCD of the numbers 10 and 21:

gcd(21, 10) = 1

GCD of numbers 16 and 23

The numbers 16 and 23 are coprime. In this case, 23 is a prime number, and 16 is a composite number.

Task: Find the GCD and LCM of numbers in the most convenient way:

a) 12 and 40; b) 9 and 40; c) 12 and 72.

5 minutes are given for the task.

What is the best way to do each exercise?

Slide breakdown.

a) It is more convenient to solve by the decomposition method into prime factors

12 = 2 2 3; 40 = 2 2 2 5

GCD(12;40)=2 2=4; LCM(12;40) = 2 2 2 3 5 = 120

b) Do the numbers 9 and 40 have common divisors? (is, 1.)

What are these numbers called? ? (Coprime.)

What is the GCD of these numbers ? (gcd(9;40) = 1)

What is the LCM of these numbers ? (LCM(9;40) = 9 40=360.)

c) What can you say about the numbers 12 and 72 ? (72 divided by 12) What rule do we know? (if one number is divisible by another, then GCD = the smallest number, and LCM - the largest)

gcd(12;72) = 12; LCM(12;72) = 72

Check the data that you have obtained with the standard that lies on the teacher's table.

FO: Evaluate themselves according to the criteria written in the standard sheet. Putting a tick next to the criterion.

7 ticks - high level

6-4 ticks - average level

1-3 ticks - low level

Fizminutka

Quickly got up, smiled,

Pulled up higher.

Well, straighten your shoulders

Raise, lower.

Turn right, turn left

Touch your hands with your knees.

Sit down, get up, sit down, get up

And they ran on the spot.

Teacher's question: Where are we already using our knowledge of GCD and LCM numbers?

When solving problems.

In front of them, on the teacher's table, there is a "Chamomile of tasks" consisting of 21 petals.

Red petal - level C tasks.

Yellow petal - tasks of level B.

Green petal - level A tasks.

	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?
	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. What is the largest number of bouquets made from these roses and how many roses of each color are in one bouquet?
	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?
	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?
	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?
	The fireplace in the room must be laid out with finishing tiles in the shape of a square. How many tiles are needed for a 195 ͯ 156 cm fireplace and what are the largest tile sizes?
	Volodya's step is 75 cm, and Katya's step is 60 cm. At what minimum distance will they both take an integer number of steps?
	For New Year's gifts, we bought 180 apples, 90 oranges and 900 sweets. All children received the same gifts. What is the largest number of identical gifts made up of these fruits and sweets?
	A garden plot measuring 54 ͯ 48 m around the perimeter must be fenced off, for this, concrete pillars must be placed at regular intervals. How many poles must be brought for the site, and at what maximum distance from each other will the poles stand?
	Find: LCM(360;252).
	For New Year's gifts, 78 chocolate bars, 156 gingerbread, 52 packs of cookies, 104 oranges and 130 apples were purchased. What is the largest number of identical gifts you can collect?
	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 flush into the box?
	Calculate GCD(720,216), LCM(720,216).
	What is the ratio of LCM (308.264) to GCM (308.264)?
	For the arrangement of the Christmas tree, they bought nuts, sweets and gingerbread - a total of 760 pieces. They took 80 more nuts than sweets, and 120 less gingerbread than nuts. What is the largest number of identical gifts for children that can be made from this stock?
	Find LCM(84,160,96),
	Find the quotient of LCM(24, 2004) divided by the GCD of the same numbers.
	Find the smallest natural number that is a multiple of 2, 3, 4, 5, 6, 7, 8, 9, 10.
	Find GCD (56, 72).
	There are books on the table, the number of which is less than 100. How many books are there if it is known that they can be tied in packs of 3, 4, and 5 pieces?
	The store brought less than 600, but more than 500 plates. When they began to lay them out in dozens, then 3 plates were not enough to reach the full number of tens, and when they began to lay them out in dozens (12 plates each), then 7 plates remained. How many plates did you bring to the store?

FD: The predominant number of petals in red indicates a high level of assimilation, yellow - an average level of assimilation and green - a low level of assimilation.