Question

Find the statement which is true about perfect squares.
A. A square number never ends with 2, 3, 6 or 7.
B. The number of zeros at the end of a perfect square is always even.
C. For any natural number n, \[{n^2}\] is the sum of first n odd natural numbers.
D. Difference between the squares of odd and even numbers is always odd.

Answer 1

Hint:- A number whose square root is also a perfect number i.e. integer is known as square numbers. Like 1, 4, 9, 16 etc. We had to check all the options because there can be more than one option correct.

Complete step-by-step answer:
Now as we know that there are different properties of a square number. So, check which of the options satisfy the properties of the square number.
Property 1:- A square number never ends with 2, 3, 7 or 8. Like 92, 83, 77 and 68 are not square numbers.
So, the square number can end with 6. Like 16 ends with 6 and it is also a square number.
So, option A is not correct.
Property 2:- Number of zeros at the end of square numbers must be even i.e. 2, 4, 6, 8 etc. Like 2500 and 3600 are square numbers while 250 and 360 are not the square numbers.
So, option B must be correct.
Property 3:- The square of a natural number n is the sum of the first n odd number. Like \[{2^2} = 4 = 1 + 3\], \[{3^2} = 9 = 1 + 3 + 5\] etc.
So, the option C must be correct i.e. for any natural number n, \[{n^2}\] must be equal to the sum of first n odd natural numbers.
Property 4:- Squares of odd numbers are always odd while the squares of even numbers are always even. Like \[{2^2} = 4\] and \[{3^2} = 9\]
So, the square of odd numbers will be odd and the square of even numbers will be even.
And as we know that odd – odd = even, even – even = even and odd – even = odd.
So, the option D will be correct.
Hence, the correct options will be B, C and D.

Note:- Whenever we come up with this type of problem then we should note that if any one of the values of the statement is false then the complete statement must be false. Like in option A, the square number never ends with 2, 3 and 7 but there was 6 also given with them and as we know that the square numbers can end with 6. So, option A will be incorrect.