Question

How do you find the number of factors of a number?

Answer 1

Hint: Factors of a number are defined as numbers or algebraic expressions that divide a given number or expression evenly. We can also say, factors are the numbers which are multiplied to get the other number. The factors of a number are defined as the number which can be multiplied to get the original number. By multiplying two factors of a number, a product is obtained, which is equal to the original number.

Complete step-by-step answer:
Let us assume N is a natural number, for which we need to find the factors. If we convert N into the product of prime numbers by prime factorization method, we can represent it as:
 \[N = {X^a} \times {Y^b} \times {Z^c}\]
where X, Y and Z are the prime numbers and a, b and c are their respective powers.
Now, the formula for the total number of factors for a given number is given by:
Total Number of Factors of a number N = \[\left( {a + 1} \right)\left( {b + 1} \right)\left( {c + 1} \right)\] .
The steps to find the factors of a number are given below in a very easy to understand way. An example is taken to make the explanation easier.
Step 1: Choose a number (say, 16)
Step 2: Write the common factors of 16 which will include \[\left( {16 \times 1} \right),\left( { - 16 \times - 1} \right),\left( {8 \times 2} \right),\left( { - 8 \times - 2} \right),\left( {4 \times 4} \right)\] and \[\left( { - 4 \times - 4} \right)\] .
Step 3: Further factor the factors until a prime number is reached. In this case, 8 can be factored further.
Step 4: Write down all the factors again. The \[\left( {8 \times 2} \right)\] will now become \[\left( {4 \times 2 \times 2} \right)\] .
Step 5: Write all the unique numbers that are obtained.
So, the factors of 16 will be 1, 2, 4, 8, 16, – 1, – 2, – 4, – 8, and – 16. Here, the positive factors of 16 are only 1, 2, 4, 8, and 16.

Note: It should be noted that factors of any number can be either positive or negative and when we multiply any two negative numbers, it results in a positive number and we know that when we multiply any two positive numbers, it results in a positive number. But normally, the factors are considered to be only positive numbers. Fractions could not be considered as factors for any number.