Question

Give the prime factorisation of the following number: 420

Answer 1

Hint: To find the prime factors, we will divide the number by the lowest prime number. To know if the number is divisible or not, we will have to make use of divisibility tests of various prime numbers. We will repeat this until the number is no longer divisible by that prime number. Then we will check if the number is divisible by the second smallest prime number or not. If it is, we will carry out the division with the next prime number and if it is not, we will move to the next prime number and in this way, we will repeat the process until the number is completely factored with all the factors as prime numbers.

Complete step by step answer:
We are supposed to find the prime factors of the number 420.
First of all, we will check whether the given number is divisible by the smallest prime number (2) or not.
The digit at the unit’s place of 420 is 0, hence the number 420 is divisible by 2.
We know that 2 times 210 is 420.
Therefore, 420 = 2 $\times $ 210.
The next number is 210, which again has a 0 at the unit’s place and thus, is divisible by 2.
We know that 2 times 105 is 210.
Therefore, 420 = 2 $\times $ 2 $\times $ 105.
105 has 5 in the unit’s place, which is not divisible by 2. 1 + 0 +5 = 6. Since the sum of the digits of the number is divisible by 3, the number is divisible by 3.
We know that 3 times 35 is 105.
Therefore, 420 = 2 $\times $ 2 $\times $ 3 $\times $ 35.
Now, 35 is not divisible by 3. The next prime number with us is 5. 35 is divisible by 5.
Therefore, 420 = 2 $\times $ 2 $\times $ 3 $\times $ 5 $\times $ 7.
7 is a prime number and only divisible by itself or 1.
Hence, the prime factorisation of 420 is 2 $\times $ 2 $\times $ 3 $\times $ 5 $\times $ 7 $\times $ 1.

Note: The above method is textual with deep explanation. Students can also solve with the help of long division method and write in compact form as follows:
 $\begin{align}
  & 2\left| \!{\underline {\,
  420 \,}} \right. \\
 & 2\left| \!{\underline {\,
  210 \,}} \right. \\
 & 3\left| \!{\underline {\,
  105 \,}} \right. \\
 & 5\left| \!{\underline {\,
  35 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7 \,}} \right. \\
 & \ \ 1
\end{align}$