Square Number
Basic
Video tutorial
Lecture Notes
Properties of square numbers:
- The ending digit of a square number can only be $0$, $1$, $4$, $5$, $6$, and $9$.
- If $n$ is even, then $n^2$ must be multiple of $4$.
- If $n$ is odd, then the remainder when $n^2$ is divided by $4$ or $8$ must be one.
- The sum of two squares cannot be in the form of $4K+3$. In another word, the remainder when the sum is divided by $4$ cannot be $3$.
- A square number will have an odd number of divisors. On the other hand, if an integer has an odd number of divisors, then it must be a square number.
Please refer to the video or the attached PDF for more fun facts of square numbers.
Attachments
Examples
If a square number's tens digit is $7$, what is its units digit?(4141)
How many terms in this sequence are squares? $$1, 11, 111, 1111, \cdots $$(4147)
Show that if $n^2$ is a square number, then $n^2\equiv 0, 1, 4, 9\pmod{16}$. In plain English, this means that the remainder can only be $0$, $1$, $4$ or $9$ when a square number is divided by $16$.(4144)
Find the number of integer pairs $(x, y)$ such that $x^2 + y^2 = 2019$.(4140)
Solve the following equation in positive integers: $3\times (5x + 1)=y^2$(2365)