This online perfect square calculator tells you whether or not any given number is a perfect square. Enter any non negative integer number in the input field of the calculator and get the answer.
Perfect Square
In mathematics, a square number or perfect square is an integer that is the product of two equal integers. If \(N\) is an integer, then \({ N }^{ 2 }\) is a perfect square. Because of this definition, perfect squares are always non-negative integers.
Some properties regarding perfect squares are as follows:
1. Perfect squares cannot have a units digit of \(2, 3,\) or \(7\).
2. The square of an even number is even and the square of an odd number is odd.
3. All odd squares are of the form \(4n+1\), hence all odd numbers of the form \(4n+3\), where \(n\) is a positive integer, are not perfect squares. For instance, \(361\) can be written as \(4 \times 90 + 1\), and we know \(361 = 19^2\). However, \(843\) is not a perfect square since \(29^{2}=841\) and \(30^{2} = 900\); it can be expressed as \(4 \times 210 + 3\).
4. All even numbers of the form \(4n + 2\), where \(n\) is a positive integer, are not perfect squares.
5. All even squares are divisible by \(4\).
6. The difference of \(2\) odd squares is a multiple of \(8\). For example, \(15^{2} – 11^{2} =104\), which is \(8 \times 13\).
7. The sum of the first \(n\) odd numbers is in fact \(n^2\). For example, \(1+3+5+7+9+11= 36\). Here, there are \(6\) odd numbers, so we can find the sum as just \(n^2=6^2\).
8. The sum of the first \(n\) perfect squares \(1^{2} + 2^{2} + 3^{2} +\cdots+n^{2}\) is given by \(\frac{n(n+1)(2n+1)}{6}\).
9. If \(p\) divides \(a^{2}\), then \(p\) divides \(a\) as well (Euclid’s theorem). From this, we can say that a number is a perfect square if its prime factorization contains all primes raised to some even power.
10. Given two positive integers \(K\) and \(m\), if \(K^2-m\) is the square of an integer \(n\), then \(K-n\) divides \(m\).
Ending digits for squared numbers in decimal system:
If a number has units digit \(1\) or \(9\), its square will have units digit \(1\).
If a number has units digit \(2\) or \(8\), its square will have units digit \(4\).
If a number has units digit \(3\) or \(7\), its square will have units digit \(9\).
If a number has units digit \(4\) or \(6\), its square will have units digit \(6\).
If a number has units digit \(5\), its square will have units digit \(5\).
If a number has units digit \(0\), its square will have units digit \(0\).
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