The topic "Multiples" is studied in the 5th grade of a comprehensive school. Its goal is to improve the written and oral skills of mathematical calculations. In this lesson, new concepts are introduced - “multiple numbers” and “divisors”, the technique of finding divisors and multiples of a natural number, the ability to find NOCs in various ways, are developed.
This topic is very important. Knowledge of it can be applied in solving fraud examples. To do this, find the common denominator by calculating the smallest common multiple (LCL).
A multiple of A is an integer that is divisible by A without a remainder.
18: 2 = 9
Each natural number has an infinite number of multiples of it. It is considered the smallest. The multiple cannot be less than the number itself.
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 by the second. If 125 is divisible by 5 without a remainder, then the answer is yes.
All natural numbers can be divided by 1. Multiple is a divisor for itself.
As we know, numbers in division are called "dividend", "divisor", "quotient".
27: 9 = 3,
where 27 is the dividend, 9 is the divisor, 3 is the quotient.
Numbers that are multiples of 2 are those that, when divided by two, do not form a remainder. These include all even ones.
Numbers that are multiples of 3 are those that are divisible by 3 (3, 6, 9, 12, 15 ...).
For example, 72. This number is a multiple of 3 because it is divided by 3 without a remainder (as you know, a number is divided by 3 without a 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 (remainder 3)
Answer: it is not, as there is a remainder.
A common multiple of two or more integers is one that is divisible by these numbers without a remainder.
K (8) = 8, 16, 24 ...
K (6) = 6, 12, 18, 24 ...
K (6.8) = 24
NOC (the smallest common multiple) is found in the following way.
For each number, it is necessary to write out multiple numbers in a line separately, up to finding the same one.
NOC (5, 6) = 30.
This method is applicable for small numbers.
When calculating the NOC, special cases are encountered.
1. If it is necessary to find the common multiple for 2 numbers (for example, 80 and 20), where one of them (80) is divisible by the 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 we can say that their NOC is the product of these two numbers.
NOC (6, 7) = 42.
Consider the last example. 6 and 7 with respect to 42 are divisors. They divide a multiple without a remainder.
42: 7 = 6
42: 6 = 7
In this example, 6 and 7 are paired dividers. Their product is equal to the most multiple (42).
6x7 = 42
A number is called prime if it is divided only by itself or by 1 (3: 1 = 3; 3: 3 = 1). The rest are called compound.
In another example, you need to determine if 9 is a divisor with respect to 42.
42: 9 = 4 (remainder 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 by which the natural numbers are divided, and the multiple itself is divided by this number.
The greatest common divisor of numbers a and b , multiplied by their least multiple, will give the product of the numbers a and b themselves.
Namely: GCD (a, b) x NOC (a, b) = a x b.
The common multiple numbers for more complex numbers are found in the following way.
For example, find NOCs for 168, 180, 3024.
We decompose these numbers into prime factors and write them in the form of a product of degrees:
168 = 2³3¹7¹
180 = 2²3²5¹
3024 = 2⁴3³7¹
Next, we write out all the presented bases of degrees with the highest rates and multiply them:
2⁴3³5¹7¹ = 15120
NOC (168, 180, 3024) = 15120.