Perfect Squares in Math

Perfect Squares

Definition

• A perfect square is a number that can be expressed as the square of an integer
• Example: 16 is a perfect square because it can be expressed as 4²

Properties

• The square of an integer is always a perfect square
• The square root of a perfect square is always an integer
• Perfect squares always end in 0, 1, 4, 5, 6, or 9

Examples

• 1, 4, 9, 16, 25, 36, 49, 64, ...
• These numbers can be expressed as 1², 2², 3², 4², 5², 6², 7², 8², ...

Identifying Perfect Squares

• If a number ends in 0, it is a perfect square if and only if the number formed by its last two digits is a perfect square
• If a number ends in 1, it is a perfect square if and only if the number formed by its last two digits minus 1 is divisible by 4
• If a number ends in 4, it is a perfect square if and only if the number formed by its last two digits minus 4 is divisible by 4
• If a number ends in 5, it is a perfect square if and only if the number formed by its last two digits minus 5 is divisible by 4
• If a number ends in 6, it is a perfect square if and only if the number formed by its last two digits minus 6 is divisible by 4
• If a number ends in 9, it is a perfect square if and only if the number formed by its last two digits minus 9 is divisible by 4

Perfect Squares

Definition

• A perfect square is a number that can be expressed as the square of an integer.
• Example: 16 is a perfect square because it can be expressed as 4².

Properties

• The square of an integer is always a perfect square.
• The square root of a perfect square is always an integer.
• Perfect squares always end in 0, 1, 4, 5, 6, or 9.

Examples

• 1, 4, 9, 16, 25, 36, 49, 64,... are perfect squares.
• These numbers can be expressed as 1², 2², 3², 4², 5², 6², 7², 8²,...

Identifying Perfect Squares

• If a number ends in 0, it is a perfect square if and only if the number formed by its last two digits is a perfect square.
• If a number ends in 1, it is a perfect square if and only if the number formed by its last two digits minus 1 is divisible by 4.
• If a number ends in 4, it is a perfect square if and only if the number formed by its last two digits minus 4 is divisible by 4.
• If a number ends in 5, it is a perfect square if and only if the number formed by its last two digits minus 5 is divisible by 4.
• If a number ends in 6, it is a perfect square if and only if the number formed by its last two digits minus 6 is divisible by 4.
• If a number ends in 9, it is a perfect square if and only if the number formed by its last two digits minus 9 is divisible by 4.