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
Square Numbers   27 Pages

Examples

 (4141) If a square number's tens digit is $7$, what is its units digit?

 (4147) How many terms in this sequence are squares? $$1, 11, 111, 1111, \cdots$$

 (4144) 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 $16$.

 (4140) Find the number of integer pairs $(x, y)$ such that $x^2 + y^2 = 2019$.

 (2365) Solve the following equation in positive integers: $3\times (5x + 1)=y^2$