Question

Identify the numbers which are not perfect squares: -
A.3107
B.6682
C.2260
D.924

Answer 1

Hint: A perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system. Examine both the unit’s digits and the digital roots of perfect squares to help determine whether the given number is a perfect square. As we know a perfect square can only end in a 0, 1, 4, 5, 6, or 9; this should allow us to determine whether the first of our numbers is a perfect square.

Complete step-by-step answer:
A perfect square is a number that can be expressed as the product of two equal integers. The only way to accurately calculate if a number is a perfect square is to find the factors. If you find the square root of a number and it's a whole integer, that tells you that the number is a perfect square.
i) 3107
\[ \to \]The given number, 3107 here 7 is in the list of numbers in units placed that are never perfect squares. The number 3,107 is not a perfect square.
ii) 6682
\[ \to \]The given number, 6682 here 2 is in the list of numbers in units placed that are never perfect squares. The number 6682 is not a perfect square.
iii) 2260
\[ \to \] The given number, 2260 here 0 is in the list of numbers in units placed that are perfect squares. So, we now need to obtain the digital root of the number:
\[2 + 2 + 6 + 0 = 10\]
If the answer is more than one digit, you would add each digit of the answer together again:
\[1 + 0 = 1\]
The digital root of number 2260 is 1, hence the number 1 is perfect square, hence we need to find the factors of 2260: 1, 2, 4, 5, 10, 20, 113, 226, 452, 565, 1130, 2260.
For a factor combination with equal numbers for X and Y (like \[3 \times 3\]) above. Notice there is no equal factor combination that, when multiplied together, produces the number 2,260.
Therefore, 2260 is not a perfect square.
iv) 924
\[ \to \]The given number, 924 here 4 is in the list of numbers in units placed that are perfect squares. So, we now need to obtain the digital root of the number:
\[9 + 2 + 4 = 15\]
If the answer is more than one digit, you would add each digit of the answer together again:
\[1 + 5 = 6\]
The digital root of number 924 is 6, hence the number 6 are never perfect squares. Therefore, 924 is not a perfect square.
Hence, all of the given numbers are not perfect squares.

Note: We must note that a perfect square can only end in a 0, 1, 4, 5, 6, or 9. A number is a perfect square (or a square number) if its square root is an integer; that is to say, it is the product of an integer with itself. A number that is a perfect square never ends in 2, 3, 7 or 8. If your number ends in any of those numbers, you can stop here because your number is not a perfect square.