Basic Understanding and Properties

Theory: A square number is a number that can be expressed as the product of a number with itself. If a natural number \(m\) can be written as \(m = n^2\), then \(m\) is a perfect square.

Example:

Question: Is 49 a perfect square?

Solution: \(49 = 7 \times 7 = 7^2\)
Therefore, 49 is a perfect square.

Theory: The area of a square is equal to the side multiplied by itself, which geometrically shows the concept of squaring.

Example:

Question: What is the area of a square with side length 8 cm?

Solution: \(\text{Area} = \text{side}^2 = 8^2 = 64\ \text{cm}^2\)

Theory: To find perfect squares between two numbers, square the consecutive natural numbers and list the results within that range.

Example:

Question: List the square numbers between 30 and 60.

Solution: \(6^2 = 36,\ 7^2 = 49\)
Therefore, perfect squares: 36, 49

Theory: A square number can only end in 0, 1, 4, 5, 6, or 9. If a number ends in 2, 3, 7, or 8, it cannot be a perfect square.

Example:

Question: Can 123 be a perfect square?

Solution: Unit digit = 3. Not allowed for square numbers.
So, 123 is not a perfect square.

Theory: You can reject some numbers as non-squares just by looking at their unit digit if it’s 2, 3, 7, or 8.

Example:

Question: Which of the following cannot be perfect squares: 1057, 23453, 222222?

Solution: Unit digits are 7, 3, and 2 respectively. All invalid.
Hence, none are perfect squares.

Theory: The square of an even number is even. The square of an odd number is odd.

Example:

Question: Determine whether the square of 727 is even or odd.

Solution: 727 is odd →\(727^2\) is odd.

Theory: A perfect square can only have an even number of zeros at the end.

Example:

Question: Is 64000 a perfect square?

Solution: It ends with 3 zeros (odd number) → Not a perfect square.

Theory: Certain patterns such as 111² = 12321 help identify squares using digit symmetry.

Example:

Question: What  is \(1111^2\)?

Solution: \(1111^2\) = 1234321

Theory: The sum of the first \(n\) odd numbers is equal to \(n^2\).

Example:

Question: What is the sum of the first 6 odd numbers?

Solution: 1+3+5+7+9+11 = 36 = \(6^2\)

Theory: There are always \(2n\) non-square numbers between \(n^2\) and \((n+1)^2\).

Example:

Question: How many numbers are between \(12^2\) and \(13^2\)?

Solution: \(12^2\) = 144, \(13^2\) = 169
Total number  = 169 – 144 – 1 = 24

Theory: Squares of even numbers are even, and squares of odd numbers are odd. Their last digits follow patterns.

Example:

Question: What is the units digit of \(34^2\)?

Solution: \(34^2\) = 1156 ⇒ Unit digit = 6

Theory: Odd square numbers can be represented as the sum of two consecutive natural numbers:
\(n^2 = \left(\frac{n^2 – 1}{2}\right) + \left(\frac{n^2 + 1}{2}\right)\)

Example:

Question: Write \(11^2\) as a sum of two consecutive integers.

Solution: \(11^2\) = 121 = 60 + 61

Theory: The product of two numbers equidistant from a number \(a\) is: \((a – 1)(a + 1) = a^2 – 1\)

Example:

Question: Show that \(13 \times 15 = 14^2 – 1\)

Solution: \(13 \times 15 = 195,\ 14^2 = 196 \Rightarrow 196 – 1 = 195\)

Theory: Squares of numbers like 1, 11, 111… create symmetric digit patterns.

Example:

Question: Find \(11111^2\)

Solution: \(11111^2\) = 123454321

Theory: Special squares like \(66\ldots672\) follow unique symmetrical patterns in the square result.

Example:

Question: Predict the square of 666672

Solution: \(666672^2\) = 4444488889