Question

Write all factors of the following numbers.
i. 36
ii. 23
iii. 96
iv. 115

Answer 1

Hint: Now to find a factor of given number we will use prime factorization method. First, divide the number by the smallest prime greater than 1. Then again consider the quotient. Now again divide the quotient with the smallest possible prime number greater than 1. Continue this process until we get 1 as quotient. Now all the numbers we used as dividends are the prime factors of the given number. Hence multiply prime numbers in all possible combinations to get all factors.

Complete step-by-step solution:
Now before finding all the factors we will first find prime factors of the given numbers.
Now to do this we will follow the method of prime factorization. First, we will divide the number by the smallest prime that is greater than 1. Now with this division, we will get a quotient. Now again divide the quotient with the smallest possible prime number that is greater than 1. Continue this process until we get 1 as quotient. Now to do this simply we write it in division form but we write the quotient just below the given number and continue the process.
Now let us understand this method by an example.
Let us say we want to find prime factors of 6.
Now the smallest prime that divides 6 is 2.
Hence let us divide 6 by 2 and write the quotient under 2
$\begin{align}
  & 2|6 \\
 & \text{ }\!\!|\!\!\text{ 3} \\
\end{align}$
Now we got quotient as 3. Now smallest prime number that divides 3 is 3 itself.
Hence we divide 3 by 3 and write quotient under the divisor 3.
$\begin{align}
  & 2|6 \\
 & 3|3 \\
 & \text{ }\!\!|\!\!\text{ 1} \\
\end{align}$
Since we got 1 as quotient we will stop the process. Hence prime factors of 6 is $2 \times 3$
Now consider 36.
Now let us use prime factorization method to find prime factors of 36
$\begin{align}
  & \text{2 }\!\!|\!\!\text{ }\overline{\text{36}} \\
 & \text{2 }\!\!|\!\!\text{ 18} \\
 & \text{3 }\!\!|\!\!\text{ 9} \\
 & \text{3 }\!\!|\!\!\text{ 3} \\
 & \text{1 }\!\!|\!\!\text{ 1} \\
\end{align}$
Hence the prime factorization of 36 is $2 \times 2 \times 3 \times 3 .$
Now we have $2 \times 2 = 4, 2 \times 3 = 6, 3 \times 3 = 9, 3 \times 2 \times 2 = 12, 2 \times 3 \times 3 = 18$
Also, note that 1 is also a factor of 36, and 36 is also a factor of 36.
Hence all factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36
Now consider 23.
Now we have that 23 is a prime number. Hence the only number that divides 23 is 1 and 23 itself.
Now consider 96.
Now let us use prime factorization method to find prime factors of 96
\[\begin{align}
  &\text{2 }\!\!|\!\!\text{ }\overline{\text{96}} \\
 & \text{2 }\!\!|\!\!\text{ 48} \\
 & \text{2 }\!\!|\!\!\text{ 24} \\
 & \text{2 }\!\!|\!\!\text{ 12} \\
 & \text{2 }\!\!|\!\!\text{ 06} \\
 & \text{3 }\!\!|\!\!\text{ 3} \\
 & \text{1 }\!\!|\!\!\text{ 1} \\
\end{align}\]
Hence the prime factorization of 96 is $2 \times 2 \times 2 \times 2 \times 2 \times 3.$
Now we have $2 \times 2 = 4, 2 \times 3 = 6, 2 \times 2 \times 2 = 8, 2 \times 2 \times 3 = 12, 2 \times 2 \times 2 \times 2 = 16, 2 \times 2 \times 2 \times 3 = 24, 2 \times 2 \times 2 \times 2 \times 2 = 32, 2 \times 2 \times 2 \times 2 \times 3 =48. $
Now again 96 and 1 are also factors of 96. Hence we get that the factors of 96 are 2, 3, 4, 8, 12, 16, 24, 32, 48.
Now let us consider the last option which is 115.
We know that 115 is not divisible by 2, 3, and 4.
But it is divisible by 5.
Hence we will use prime factorization method and start with dividing 115 by 5.
$\begin{align}
  & 05|115 \\
 & 23|23 \\
 & 01|1 \\
\end{align}$
Hence we get prime factorization of 115 is $5 \times 23.$
Hence all the factors of 115 are 1, 5, 23, 115.

Note: Note that is the number is small we can always check the factors by dividing each natural number less than our given number. If the remainder is 0 then the natural number is a factor if not then we will have to proceed to the next number. For checking factors one must be comfortable with multiplication tables too. Also, remember 1 and the number itself are always factors of the number.