Factors of a Number

Introduction

A factor in mathematics is an integer that evenly divides another number by itself, leaving no residue. We encounter factors and multiples regularly. In the tutorial, we will discuss more Prime and composite number factors, HCF of at least two numbers and we shall solve a few related examples.

What is Factor?

In mathematics, a factor is a divisor of a given integer that divides it exactly, leaving no leftovers. For instance, they are employed while handling money, sorting objects into groups, looking for patterns in numbers, resolving ratios, and expanding or contracting fractions. We can employ a variety of techniques, including the division method and the multiplication method, to determine the factors of a given integer.

For example, 2 and 4 is factors of 12 because 12 ÷ 2 = 6 exactly and 12 ÷ 4 = 3 exactly. The other factors of 12 is 1, 3, 6, and 12. Therefore the factor of 12 is 1, 2, 3, 4, 6, and 12.

Characteristics of Factor

A number's factors have the following characteristics −

• A number has a finite number of factors.

• A number's factor will never be more than or equal to the provided number.

• Every number contains at least two factors, 1 and the actual number, except of 0 and 1.

• Finding a number's factors involves using division and multiplication operations.

Method to find Factor

If you want to determine the factors of any integer, you can do so in one of two ways −

• Factors using multiplication − If we want to determine the factors of a number, let's say N, we should multiply the two numbers in different ways so that the result is equal to N. The factors is all the separate numbers whose addition produces a product equal to N.

  For example, factor of 18 using above method as,

  1×18=18, 2×9=18, 3×6=18

  Therefore factors of 18 are 1, 2, 3, 6, 9, and 18.

• Factors using Division − Let's say that we are asked to determine the factors of a certain integer N. Use a number smaller than N. Now subtract N from the numbers so that the quotient that results is a whole number. Thus, without any repetition, all of the quotients and divisors involved in this situation become the factors of N.

  For example, the factor of 18 using the above method as,

  18÷1=18, 18 ÷2=9, 18÷3=6

  Therefore factors of 18 are 1, 2, 3, 6, 9, and 18.

Prime and Composite Numbers

There are two different kinds of numbers in mathematics that depend on the factors of numbers. Prime numbers and composite numbers are these two categories of numbers.

The two factors, one and itself are the only factors that prime numbers maintain. The number can only be divided by itself and by one, according to this implication. Except for 2, every prime number is an odd number. For example: 2,3,5,7,11 and so on.

The term composite number refers to an integer that has at least one factor other than 1 and can be calculated by multiplying the two smallest positive numbers together. Prime numbers and units are not the same thing as composite numbers. A composite number is any non-prime number. For example: 4,6,8,9,10 and so on.

There are two types of composite numbers, odd and even composite numbers. An odd composite number is any odd integer that is not a prime number. For instance, the composite numbers 9, 15, 21, 25, 27, and 33 are odd. An even composite number is any even integer that is not a prime number. For instance, the composite numbers 4, 6, 8, 10, and 12 are all even.

What is HCF?

The highest common factor, or HCF, is the common factor between any two or more numbers. The greatest common factor (GCF) or greatest common divisor are alternate names for it (GCD). For instance, the HCF of 2 and 4 is 2, as 2 is the number that both 2 and 4 shares.

By using the division method or the prime factorization method, we can determine the HCF of a set of supplied natural numbers. Given numbers are expressed as the product of prime factors when using the prime factorization method. In contrast, when using the division method, the given integers are divided by the least common multiple until there is zero remainders.

Solved examples

1) Find HCF of 45 and 27 using prime factorization method.

Answer: Write each number as a product of prime factors in order to determine the HCF of 45 and 27.

$$\mathrm{45=3×3×5=3^2×5}$$

$$\mathrm{27=3×3=3^2}$$

Now multiply all of the usual prime factors by the smallest power. Here, the lowest power of 2 and the only common prime factor is 3.

Therefore the HCF of 45 and 27 is 3²=9.

2) Find HCF of 36, 54, and 24 using the division method.

Answer: Write each number as a product of prime factors in order to determine the HCF of 36, 54 and 24.

$$\mathrm{36=2×2×3×3=2^2×3^2}$$

$$\mathrm{54=2×3×3×3=2×3^3}$$

$$\mathrm{24=2×2×2×3=2^3×3}$$

Therefore the HCF of 36, 54 and 24 is 2×3=6.

Conclusion

A factor is a divisor of a given integer that divides it exactly, leaving no leftovers. The two factors, one and itself are the only factors that prime numbers maintain. The term composite number refers to an integer that has at least one factor other than 1 and can be calculated by multiplying the two smallest positive numbers together. Prime numbers and units are not the same thing as composite numbers.

FAQs

1.What are the characteristic of prime numbers?

The characteristic of prime numbers are as follow −

• The only even prime number is two.

• The numbers 0 and 1 are neither prime nor composite.

• Not a single prime number greater than 5 ends with a 5.

• It is non zero whole number.

• Prime numbers cannot be divided by any number 1 and itself.

2.What are the characteristic of composite numbers?

The characteristic of prime numbers are as follow −

• A number 4 is smallest composite number.

• The numbers 0 and 1 are neither prime nor composite.

• All even number except 2 is even.

• Every composite number can be written as product of 2 or more prime numbers.

• Sphenic number is composite number with three different factors.

3.What is LCM?

The term "LCM" refers to the least or smallest common multiple of any two or more specified natural numbers. The LCM of 10, 15, and 20 equals 60, for instance.

4.What are Coprime numbers?

The two numbers known as coprime-numbers, mutually primes, or comparatively primes have only one element in common, which is 1.

For example, the factors of 14 are 1, 2 and 7 and the factors of 15 are 1, 3 and 5. here common factor is 1. Therefore, 14 and 15 are coprime numbers but for 21, the factors of 21 are 1, 3, and 7 then here 21 is neither coprime with 14 nor with 15.

5.What is prime factoriziation?

A number can be expressed as the product of its prime factors by prime factorization. For example, the prime factorization of 80 = 2 × 2 × 2 × 2 × 5. In this case 2 and 5 are the prime factors of 80. The factors that are prime numbers are known as prime factors.