5.2-Properties of Square Numbers

A square number (also called a perfect square) is a number that can be expressed as the product of an integer multiplied by itself. In other words, a square number is the square of an integer. The general form of a square number is $n^2$, where $n$ is an integer.

Some of the key properties of square numbers are discussed below:

• Property 1: A square number is always non-negative.
  Any square number is always greater than or equal to zero. For example, the square of any positive integer or zero is always non-negative. For instance, $3^2 = 9$, $0^2 = 0$, and $(-4)^2 = 16$.
• Property 2: A square number has an odd or even root.
  The square of an even number is always even, and the square of an odd number is always odd. For example, $(2)^2 = 4$ (even) and $(3)^2 = 9$ (odd). In general: - If $n$ is even, then $n^2$ is even. - If $n$ is odd, then $n^2$ is odd.
• Property 3: The last digit of a square number follows a specific pattern.
  The last digit of a square number can only be one of the following: $0, 1, 4, 5, 6, 9$. It cannot be any other digit. For example: - $4^2 = 16$ (last digit is 6) - $7^2 = 49$ (last digit is 9) - $12^2 = 144$ (last digit is 4)
• Property 4: The difference between two consecutive square numbers increases in a linear manner.
  The difference between the square of any two consecutive numbers increases by a constant value of $2n + 1$. For example: - The difference between $3^2 = 9$ and $4^2 = 16$ is $16 - 9 = 7$ (which is $2 \times 3 + 1$). - The difference between $5^2 = 25$ and $6^2 = 36$ is $36 - 25 = 11$ (which is $2 \times 5 + 1$).
• Property 5: The sum of the first $n$ odd numbers is always a perfect square.
  The sum of the first $n$ odd numbers is equal to $n^2$. This can be written as: $$ 1 + 3 + 5 + 7 + \dots + (2n - 1) = n^2 $$ For example: - The sum of the first 3 odd numbers: $1 + 3 + 5 = 9 = 3^2$. - The sum of the first 4 odd numbers: $1 + 3 + 5 + 7 = 16 = 4^2$.
• Property 6: Square numbers cannot end with 2, 3, 7, or 8.
  A square number cannot have a last digit of 2, 3, 7, or 8. For example, numbers like 32, 53, 72, and 88 are not squares of any integer. Thus, the square of a number cannot end with these digits.
• Property 7: A number whose prime factorization contains pairs of primes is a square number.
  A number is a perfect square if and only if each prime factor in its prime factorization appears an even number of times. For example: - $36 = 2^2 \times 3^2$ (Both prime factors 2 and 3 have even powers, so 36 is a perfect square). - $72 = 2^3 \times 3^2$ (Here, 2 appears with an odd power, so 72 is not a perfect square).
• Property 8: The square of a negative number is always positive.
  When a negative number is squared, the result is always positive. This is because multiplying two negative numbers gives a positive product. For example: - $(-5)^2 = 25$ - $(-9)^2 = 81$