Question

For an odd number 427, the square of this odd number will be_
A. Even
B. Odd
C. Negative
D. Positive

Answer 1

Hint:For a number to be odd it should not have 2 as its factor. Or more simply we can say that the last digit should not be 0, 2, 4 and 8.

Complete step-by-step answer:
We are given that 427 is an odd number and we need to find whether the square of this number is even or odd.
First we will start by looking for factors of 427.
The factors can be calculated from prime factorisation. Thus, we get
\[427{\text{ }} = {\text{ }}61 \times 7\]
Therefore, the factors of 427 will be 1, 7, 61 and the number itself.
Since, two is not a factor of 427 and when we square it the same will be the case of its squared value i.e. there will be no factor equal to 2. This can be shown as
\[{427^2}{\text{ }} = {\text{ }}{61^2} \times {7^2}\]
Since, there are no factors of it in the squared value of 427 it will be odd.
Thus, it can be said that the square of an odd number is always odd.

So, the correct answer is “Option A”.

Note:Another method which can be used is calculating the square of the number and checking afterwards whether it is even or odd. This is shown as
\[{427^2}{\text{ }} = {\text{ }}{61^2} \times {7^2} = 182323\]
Since, the last digit is none of the 2, 4, 6 or 0 the number obtained is odd. This method can be tedious if the number given is large.Similar to the square, the square root of an odd number is an odd number.