Multiples are an essential concept in mathematics that relates to the relationship between numbers. In simple terms, a multiple of a number is obtained by multiplying it by any whole number. For example, if we consider the number 4, its multiples are 4, 8, 12, 16, 20, and so on.
In this article, we will explore the concept of multiples in mathematics in more detail and discuss some essential facts that every math student should know.
What is a multiple of a number?
A multiple of a number is obtained by multiplying it by any whole number. In mathematical terms, we say that a number “n” is a multiple of another number “m” if there exists an integer “k” such that “n” can be expressed as “n = k × m”.
For example, let’s consider the number 6. Its multiples are 6, 12, 18, 24, 30, and so on. We can express these multiples as follows:
- 6 = 1 × 6
- 12 = 2 × 6
- 18 = 3 × 6
- 24 = 4 × 6
- 30 = 5 × 6
From this, we can see that each multiple of 6 is obtained by multiplying it by a whole number.
How to find multiples of a number?
To find the multiples of a given number, we need to multiply it by the first few whole numbers. For example, if we want to find the multiples of 4, we can multiply it by 1, 2, 3, 4, 5, and so on. The resulting numbers will be the multiples of 4. We can express these multiples in the following way:
- 4 × 1 = 4
- 4 × 2 = 8
- 4 × 3 = 12
- 4 × 4 = 16
- 4 × 5 = 20
- 4 × 6 = 24
- 4 × 7 = 28
- 4 × 8 = 32
- 4 × 9 = 36
From this, we can see that the multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, and 36.
Properties of multiples
Multiples have several essential properties that make them useful in many areas of mathematics. Some of these properties include:
1. Closure Property
The closure property of multiples states that the multiples of a number are always integers. For example, the multiples of 5 are 5, 10, 15, 20, and so on, which are all integers.
2. Divisibility Property
The divisibility property of multiples states that if “n” is a multiple of “m”, then “m” divides “n” without a remainder. For example, 12 is a multiple of 3, and we can express it as 12 = 3 × 4. This shows that 3 divides 12 without a remainder.
3. Multiples of 1
Every number is a multiple of 1. For example, the multiples of 1 are 1, 2, 3, 4, 5, and so on.
4. Relationship between multiples and factors
Multiples and factors are related concepts in mathematics. If “n” is a multiple of “m”, then “m” is a factor of “n”. Similarly, if “p” is a factor of “q”, then “q” is a multiple of “p”. For example, if 6 is a multiple of 2, then 2 is a factor of 6. Likewise, if 15 is a multiple of 3, then 3 is a factor of 15.
5. Multiples of 0
Every number is a multiple of 0. For example, the multiples of 0 are 0, 0 × 1, 0 × 2, 0 × 3, and so on.
6. Least Common Multiple (LCM)
The least common multiple of two or more numbers is the smallest number that is a multiple of all of them. For example, the LCM of 4 and 6 is 12, since 12 is the smallest number that is a multiple of both 4 and 6.
To find the LCM of two or more numbers, we can use the following steps:
- Write the prime factorization of each number.
- Take the highest power of each prime factor.
- Multiply these highest powers together.
For example, to find the LCM of 12 and 18, we can follow these steps:
- The prime factorization of 12 is 2 × 2 × 3.
- The prime factorization of 18 is 2 × 3 × 3.
- The highest power of 2 is 2, and the highest power of 3 is 2.
- Therefore, the LCM of 12 and 18 is 2 × 2 × 3 × 3 = 36.
7. Multiples of Fractions
We can also find the multiples of fractions. To do this, we need to multiply the numerator and denominator of the fraction by the same number. For example, if we want to find the multiples of 3/4, we can multiply both the numerator and denominator by the first few whole numbers:
- 3/4 × 1 = 3/4
- 3/4 × 2 = 6/8
- 3/4 × 3 = 9/12
- 3/4 × 4 = 12/16
From this, we can see that the multiples of 3/4 are 3/4, 6/8, 9/12, 12/16, and so on.
Conclusion
In conclusion, multiples are an essential concept in mathematics that relate to the relationship between numbers. A multiple of a number is obtained by multiplying it by any whole number. We can find the multiples of a number by multiplying it by the first few whole numbers.
Multiples have several important properties, including the closure property, divisibility property, and the relationship between multiples and factors.
We can also find the LCM of two or more numbers using the prime factorization method. By understanding these concepts, we can solve many mathematical problems that involve multiples.