Question

252  be expressed as a product of prime numbers as:-

a)\[2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }7\]

b)\[3\text{ }\times \text{ }3\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }7\]

c) \[7\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\]

d)\[~2\text{ }\times \text{ }7\text{ }\times \text{ }3\text{ }\times \text{ }2\text{ }\times \text{ }3\]

Answer 1

Hint: We will have to do the prime factorization of 252 for solving this question.

Let us first know about Prime Factorization.

Prime factorization: The method of prime factorization is used to “break down” or express a given number as a product of prime numbers.

Let us take an example. Say, 468.

$   \ \underline{2\left| 468 \right.} $

$  \ \underline{2\left| 234 \right.} $

$  \underline{\ 3\left| 117 \right.} $

$  \underline{\ 3\left| 39 \right.} $

$  \underline{13\left| 13 \right.} $

$  \ \ \ \left| 1 \right. $

So, \[468\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }13\]

So, this is how we can do the prime factorization of numbers.

Complete step by step answer:

Let us now solve the question.

Prime Factorization of 252

$   \ \underline{2\left| 252 \right.} $

$ \ \underline{2\left| 126 \right.} $

$ \underline{\ 3\left| 63 \right.} $

$  \underline{\ 3\left| 21 \right.} $

$  \ \underline{7\left| 7 \right.} $

$ \ \ \ \left| 1 \right. $

So, \[252\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }7\]

We can observe that when we multiply the prime factors, then we will get 252.

\[2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }7\text{ }=\text{ }4\text{ }\times \text{ }9\text{ }\times \text{ }7\text{ }=\text{ }36\text{ }\times \text{ }7\text{ }=\text{ }252\]

So, the correct answer of this question is (a) \[2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }7\] .

We can also write \[2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }7\] as \[{{2}^{2}}\times \text{ }{{3}^{2}}\times \text{ }7\] to make it look more compact.

Note:

One must do all the calculations very carefully for solving this question. A mistake in one step leads to the wrong answer.

Also, not only in this question, one must be very careful while doing such questions as if there is any mistake in the calculations, the answer can come out to be wrong.