Question

Define factor. Give four examples.

Answer 1

Hint: First we will understand the meaning of the term ‘factor’ of a number. Now, we will consider some numbers whose factors we will find as our examples. To find the factors of a number we will use the prime factorization method and write the numbers as the product of their prime factors. Further, we will group these prime factors to find the different composite factors.

Complete step by step answer:
Here we have been asked to define the term ‘factor’ and give four examples also. First let us understand about the term ‘factor’.
Now, in mathematics the term ‘factor’ (x) of a number (y) means x divides y completely without leaving any remainder. A number can have many factors which are obtained by grouping their prime factors. If a positive integer has only two factors then that integer is called a prime number and if a positive integer has more than two factors then that integer is called a composite number. Let us take some examples: -
(1) Consider the number 5. It is a prime number so it has only two factors, they are 1 and the number itself 5.
(2) Consider the number 15. Now, 15 is a composite number so it can be written as the product of its prime factors as $15=3\times 5$. Therefore, we can say that in addition to 1 and 15 the given number has two more factors 3 and 5.
(3) Consider the number 56. It can be written as $56=2\times 2\times 2\times 7$, so 56 can be divided by 2, 4, 7, 8, 14 and 28 without leaving any remainder, so they are the factors of 56 in addition to 1 and 56.
(4) Consider the number 100. It can be written as $100=2\times 2\times 5\times 5$, so grouping the factors properly we can conclude that 100 can be divided by 2, 4, 5, 10, 20, 25 and 50 in addition to 1 and 100. Therefore, the listed numbers are the factors of 100.

Note: Note that 1 is a factor of every number and every number is a factor of itself. If you are asked to find the number of factors (including both prime and composite) then write the given number in the form ${{a}^{m}}\times {{b}^{n}}\times {{c}^{p}}...$ and so on. Here a, b, c…. are the different prime factors and m, n, o… are their exponents. Use the formula for the number of prime factors given as $\left( m+1 \right)\left( n+1 \right)\left( p+1 \right)....$ to get the answer.