Greatest common divisor. Mutually prime numbers. Least common multiple. What numbers are relatively prime? What are the properties of coprime numbers? Properties of coprime numbers

The natural numbers a and b are called mutually prime, if their greatest common divisor is 1 (GCD(a; b) = 1). In other words, if the numbers a and b have no common factors 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) = 1. For example, (23, 30) = 1. This notation is, as it were, a shorthand notation for the greatest common divisor of two numbers (GCD(23, 30) = 1), and says that their greatest common divisor is 1.

Two adjacent natural numbers will always be relatively prime. For example, 15 and 16 are a pair of relatively prime numbers, just like 16 and 17. This is easy to understand if you 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 divisible by an integer, i.e., 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.

From the definition of coprime numbers and prime numbers it also follows that different prime numbers are always coprime. After all, the divisors of any prime number are only 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 (this means coprime), therefore 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 ab. This can be written like this: 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 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. We check: 68 ÷ 17 = 4, i.e. it is divisible by a whole, which means 68 is a multiple 17.

Typically there are more properties than are listed here. In addition, the properties of coprime numbers are formulated in different ways. It may also be necessary to prove these properties (in this case no proof is given).

The greatest common divisor of coprime numbers is always one.

Examples of coprime number nodes.

GCD of numbers 11 and 7

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

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

gcd(11, 7) = 1

GCD of numbers 11 and 15

The numbers 11 and 15 are relatively prime. Moreover, 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. 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. Moreover, both the number 10 and the number 21 are composite.

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

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

As you can see, the only common factor of the numbers 10 and 21 is the number 1. 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 relatively prime. Moreover, 23 is a prime number, and 16 is a composite number.

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

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

The task is given 5 minutes.

What is the most convenient way to solve each exercise?

Analysis by slide.

a) It is more convenient to solve by factoring 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 factors? (there is, 1.)

What are these numbers called? ? (mutually prime.)

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

What is the LCM of these numbers? ? (NOC(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 you got with the standard that lies on the teacher’s desk.

FO: They evaluate themselves according to the criteria written on the standard sheet. By checking the box next to the criterion.

7 ticks – high level

6-4 ticks – average level

1-3 ticks – low level

Fizminutka

They quickly stood up, smiled,

They pulled themselves higher and higher.

Well, straighten your shoulders,

Raise, lower.

Turn right, turn left,

Touch your hands with your knees.

They sat down, stood up, sat down, stood up,

And they ran on the spot.

Teacher question: Where do we already use our knowledge of GCD and LCA of numbers?

When solving problems.

In front of them on the teacher’s desk is a “Task Chamomile” consisting of 21 petals.

Red Petal – level C tasks.

Yellow petal – level B tasks.

Green petal – level A tasks.

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