Question

Express each number as a product of its prime factors:
$
  \left( i \right){\text{ 140}} \\
  \left( {ii} \right){\text{ 156}} \\
  \left( {iii} \right){\text{ 3825}} \\
 $

Answer 1

Hint: Here, we will proceed by using the prime factorisation method so that each of the given numbers can be represented as the product of the prime factors of the given number which needs to be represented in such a form.

Complete step-by-step answer:
By prime factorisation method, we can express the given numbers as a product of its prime factors
$\left( i \right)$ The number given is 140 and it can be represented as under
$140 = 2 \times 2 \times 5 \times 7$
The above representation consists of all the prime factors of 140. Clearly, all the numbers 2, 5 and 7 are prime numbers which leaves zero as remainder when the number 140 is divided by any one of these.
$\left( {ii} \right)$ The number given is 156 and it can be represented as under
$156 = 2 \times 2 \times 3 \times 13$
The above representation consists of all the prime factors of 156. Clearly, all the numbers 2, 3 and 13 are prime numbers which leaves zero as remainder when the number 156 is divided by any one of these.
$\left( {iii} \right)$ The number given is 3825 and it can be represented as under
$3825 = 3 \times 3 \times 5 \times 5 \times 17$
The above representation consists of all the prime factors of 3825. Clearly, all the numbers 3, 5 and 17 are prime numbers which leaves zero as remainder when the number 3825 is divided by any one of these.

Note: This representation is very useful in evaluating the square root, the cube root or the root of any higher degree of any number. For the square root of any number, pairs of two same prime numbers can be taken outside of the root. For the cube root of any number, pairs of three same prime numbers can be taken outside of the root.