The topic "Multiple numbers" 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 LCM in various ways.
This topic is very important. Knowledge on it can be applied when solving examples with fractions. To do this, you need to find the common denominator by calculating the least common multiple (LCM).
A multiple of A is an integer that is divisible by A without a remainder.
18:2=9
Every natural number has an infinite number of multiples of it. It is considered to be the least. A multiple cannot be less than the number itself.
Task
You need to prove that the number 125 is a multiple of the number 5. To do this, you need to divide the first number 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. A multiple is a divisor of itself.
As we know, when dividing numbers 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 numbers.
Numbers that are multiples of 3 are those that are divisible by 3 without a remainder (3, 6, 9, 12, 15…).
For example, 72. This number is a multiple of 3, because it is divisible by 3 without a remainder (as you know, a number is divisible by 3 without a remainder if the sum of its digits is divisible by 3)
sum 7+2=9; 9:3=3.
Is 11 a multiple of 4?
11:4=2 (remainder 3)
Answer: not, as there is a remainder.
A common multiple of two or more integers is one that is evenly divisible by those numbers.
K(8)=8, 16, 24…
K(6)=6, 12, 18, 24…
K(6, 8)=24
LCM (least common multiple) is found in the following way.
For each number, you must separately write multiple numbers in a line - up to finding the same.
NOK (5, 6)=30.
This method is applicable for small numbers.
There are special cases in calculating the LCM.
1. If you need to find a common multiple for 2 numbers (for example, 80 and 20), where one of them (80) is divisible by the other (20) without a remainder, then this number (80) is the smallest multiple of these two numbers.
NOK (80, 20)=80.
2. If two prime numbers do not have a common divisor, then we can say that their LCM is the product of these two numbers.
NOK (6, 7)=42.
Let's consider the last example. 6 and 7 in relation to 42 are divisors. They sharea multiple without a remainder.
42:7=6
42:6=7
In this example, 6 and 7 are pair divisors. Their product is equal to the most multiple number (42).
6х7=42
A number is called prime if it is divisible only by itself or by 1 (3:1=3; 3:3=1). The rest are called composite.
In another example, you need to determine if 9 is a divisor with respect to 42.
42:9=4 (remaining 6)
Answer: 9 is not a divisor of 42 because the answer has a remainder.
A divisor differs from a multiple in that the divisor is the number by which natural numbers are divided, and the multiple is itself divisible 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 LCM (a, b)=a x b.
Common multiples for more complex numbers are found in the following way.
For example, find the LCM for 168, 180, 3024.
These numbers are decomposed into prime factors, written as a product of powers:
168=2³x3¹x7¹
180=2²x3²x5¹
3024=2⁴x3³x7¹
Next, we write out all the presented bases of degrees with the largest exponents and multiply them:
2⁴x3³x5¹x7¹=15120
NOK (168, 180, 3024)=15120.
