Question

Without doing the calculation, find the numbers which are surely not perfect squares.
I.153
II.257
III.408
IV.441

Answer 1

Hint: The number resulting when a number is multiplied by itself is called a square of a number. In general, it is the product of the number by itself. To check whether the number is a perfect square, we factorize the number, and their same prime numbers are paired in the case of a square.
But in this question, we have to check the numbers whether they are a perfect squared number or not without doing calculations; hence, for this question, we will approach by checking the unit digit of the given numbers.

Complete step-by-step answer:
We know if the unit digit of the given number is 2, 3, 7, 8, then the given number is not a perfect square, whereas if the given numbers end with 0, 1, 4, 5, 6, and 9, then the number could be a perfect square.
I.153
The unit digit of the number is 3; hence we can say the number is not a perfect square number.
II.257
The unit digit of the number is 7 hence we can say the number is not a perfect square number.
III.408
The unit digit of the number is 8; hence we can say the number is not a perfect square number.
IV.441
The unit digit of the number is 1, so we can say that the number 441 might be a perfect square. Moving further, we need to see that 400 is the nearest number to 441 which is the square of the number 20, so we have to check that 21 is the square root of 441 or not.
$\Rightarrow$ Multiply 21 two times such that $21 \times 21 = 441$.
Hence, we can say that 441 is a perfect square.
Hence from the above observations, we can conclude that the number 153, 257, and 408 are not a perfect square number.

Note: This method of checking the numbers to be a perfect square or not does depend upon the number of digits of the number, whatever might be the number of digits the given number has just checked for the unit place digits.