Divisors and multiples

The topic "Multiple numbers" is studied in the 5th grade of the general education school. Its goal is to improve the written and oral skills of mathematical calculations. In this lesson, new concepts are introduced - "multiple numbers" and "dividers", the technique of finding divisors and multiple natural numbers is worked out, the ability to find NOCs in various ways.

This topic is very important. Knowledge of it can be applied to solving examples with fractions. To do this, it is necessary to find a common denominator by calculating the smallest common multiple (NOC).

A multiple A is an integer that is divisible by A without remainder.

18: 2 = 9

Each natural number has an infinite number of multiple numbers. It is considered to be the smallest. Multiple can not be less than the number itself.

A task

It is necessary to prove that the number 125 is a multiple of the number 5. For this, the first number must be divided into the second. If 125 is divisible by 5 without remainder, then the answer is positive.

All natural numbers can be divided by 1. A multiple is a divisor for itself.

As we know, the numbers in the division are called "dividend", "divisor", "private".

27: 9 = 3,

Where 27 is a dividend, 9 is a divisor, and 3 is a quotient.

The numbers that are multiples of 2 are those that, when divided by two, do not form a remainder. They are all even.

The numbers that are multiples of 3 are those that divide without a remainder into 3 (3, 6, 9, 12, 15 ...).

For example, 72. This number is a multiple of the number 3, because it is divisible by 3 without remainder (as is known, the number is divided by 3 without remainder if the sum of its digits is divided by 3)

The sum of 7 + 2 = 9; 9: 3 = 3.

Is the number 11 a multiple of 4?

11: 4 = 2 (balance 3)

Answer: it is not, as there is a remainder.

A common multiple of two or more integers is one that is divided into these numbers without a remainder.

K (8) = 8, 16, 24 ...

K (6) = 6, 12, 18, 24 ...

K (6.8) = 24

The LCM (the least common multiple) is found in the following way.

For each number, it is necessary to separately write out multiple numbers in a row - up to finding the same number.

NOC (5, 6) = 30.

This method is applicable for small numbers.

When calculating NOC, there are special cases.

1. If it is necessary to find a common multiple for 2 numbers (for example, 80 and 20), where one of them (80) is divided without remainder by another (20), then this number (80) is the smallest multiple of these two numbers.

NOC (80, 20) = 80.

2. If two prime numbers do not have a common divisor, then one can say that their LCM is a product of these two numbers.

NOC (6, 7) = 42.

Let's consider the last example. 6 and 7 with respect to 42 are divisors. They divide a multiple number without a remainder.

42: 7 = 6

42: 6 = 7

In this example, 6 and 7 are paired divisors. Their product is equal to the most multiple of (42).

6x7 = 42

A number is said to be simple if it divides itself by itself or by 1 (3: 1 = 3; 3: 3 = 1). The rest are called composite.

In another example, you need to determine whether 9 is a divisor relative to 42.

42: 9 = 4 (balance 6)

Answer: 9 is not a divisor of 42, because there is a remainder in the answer.

The divisor differs from the multiple in that the divisor is that number divided by natural numbers, and the multiple itself is divided by this number.

The greatest common divisor of a and b , multiplied by their least multiple, yields the product of the numbers a and b themselves.

Namely: GCD (a, b) x LCM (a, b) = a x b.

Common multiple numbers for more complex numbers are found in the following way.

For example, to find an LCM for 168, 180, 3024.

We decompose these numbers into prime factors, we write them as products of powers:

168 = 2³х3¹х7¹

180 = 2²х3²х5¹

3024 = 2⁴х3³х7¹

Further, we write out all the presented bases of degrees with the largest indicators and multiply them:

2⁴х3³х5¹х7¹ = 15120

NOC (168, 180, 3024) = 15120.