4 is an even number that is a multiple of 2.
Multiple is an adjective that is used in mathematics and grammar , coming from the word multiplus . In mathematics, it is the number or quantity that contains another several times exactly .
An integer r is a multiple of an integer s when there is another natural number that, multiplied by s , gives us r as a result . For example: 12 is a multiple of 3 since 3 x 4 = 12 . We see then that if we multiply 3 by 4 , we have the result 12 , which means that 12 is a multiple of 3 .
If we want to know if one number is a multiple of another, we must perform a division operation between both. When the quotient is an integer (and, therefore, the remainder or remainder of the operation is 0 ), we are dealing with a number that is a multiple of the other. Going back to our previous example, 12 / 3 = 4 (so the product of 3 and 4 is 12 ).
Properties of multiples
The set of multiples of a natural number is infinite . In other words, there are as many multiples of a number as there are natural numbers . The multiples of 3 are {3, 6, 9, 12, 15. 18, 21…} . Let's look at the properties of multiples below:
- All integers are multiples of 1, since any number can be obtained by multiplying by 1 (18 x 1 = 18, 134 x 1 = 134).
- If a number a is a multiple of b , it is also given that b is a divisor of a (since 33 is a multiple of 11, 11 is a divisor of 33, since 33 / 3 = 11).
- 0 (zero) is a multiple of any number, that is, when you multiply it by any value you get 0.
- By adding different multiples of a given number, another multiple of it is obtained.
- Similar to the previous point, the difference of two multiples of any number results in a third multiple.
- If a is a multiple of b and this is a multiple of c , then a is a multiple of c .
- If a is a multiple of b , all multiples of a are also multiples of b .
The concept of multiple is important when analyzing divisibility criteria.
The prime numbers
A prime number is one that is greater than 1 and only has two positive divisors : 1 and itself. On the other hand, a composite number has at least one other divisor than 1 and itself, which can be decomposed into factors ( factorization ).
A prime multiple is called, in this framework, the number that can be divided by a prime number with a remainder of 0. An example of a prime multiple is 210, which can be divided in this way by 3 and 7, among others.
When two positive numbers do not have a prime factor in common and, therefore, only 1 is their common divisor, they are relative primes . 6 is a relative prime number of 11 and 35, for example.
Multiples of a natural number constitute an infinite set of numbers.
Least common multiple
The smallest common multiple (LCM) is the smallest number among all positive numbers that is a multiple of at least two numbers. The LCM is usually used for subtracting or adding fractions when each fraction of the operation in question has a different denominator.
The greatest common divisor (GCD) , for its part, is the largest integer that divides two or more integers with remainder 0. It can be calculated by resorting to decomposition into prime factors.
The submultiples
A submultiple is any number that contains another number exactly (with remainder 0) a certain number of times. For example, 3 is a submultiple of 27, since it contains it 9 times (3 x 9 = 27). Seen from another perspective, for this condition to occur it is also necessary that the second number be a multiple of the first (27 is, in effect, a multiple of 3).
In the real world, in practice, multiples and submultiples have various applications , the most common being the conversion of units of measurement to facilitate the understanding of information; For example, to talk about the length of a road we usually talk in kilometers, while to express the thickness of a mobile phone, millimeters are used.
Units of measurement, multiples and submultiples
Let's look at some of the submultiples of the most used units of measurement:
- Gram : decigram, centigram and milligram.
- Liter : deciliter, centiliter and milliliter.
- Meter : decimeter, centimeter and millimeter.
As we already indicated, the use of submultiples is associated with simplicity . If we want to refer to the size of an insect, the logical thing is to talk about centimeters or millimeters and not meters. Likewise, the weight of this type of animal is expressed in milligrams, not grams.
On the other hand, let's analyze the properties of submultiples:
- 1 is a submultiple of any given number, since dividing a number by itself always gives 1.
- All numbers are submultiples of themselves, since they are obtained as a result of dividing them by 1.
- All numbers are submultiples of 0.
Other meanings
In the field of grammar, an adjective or multiple numeral noun is one whose meaning is generated from the multiplication of a quantity .
Double, triple, quadruple y quintuple son ejemplos de múltiplos: «My room is twice as big as yours», "The musician was subjected to a triple bypass".