#### Practice (93)

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Show that $2x^2 - 5y^2 = 7$ has no integer solution.

For any given positive integer $n$, prove $(n^2 +n +1)$ cannot be a perfect square.

How many positive integers not exceeding $2000$ have an odd number of factors?

Show that the difference of two squares of odd numbers must be a multiple of $8$.

Show that there exists an infinite sequence of positive integers $a_1, a_2, \cdots$ such that $$S_n=a_1^2 + a_2^2 + \cdots + a_n^2$$ is square for any positive integer $n$.

Let $k$ be a positive integer, show that $(4k+3)$ cannot be a square number.

How many numbers in this series are squares? $$1, 14, 144, 1444, 14444, \cdots$$

Find all positive integer $n$ such that $n$ is a square and its last four digits are the same.

Solve the following equation in positive integers: $15x - 35y + 3 = z^2$

Find a four-digit square number whose first two digits are the same and the last two digits are the same too.

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

Find all pairs of integers $(x, y)$ such that $5\times (x^2 + 3)= y^2$.

If we arrange all the square numbers ascendingly as a queue: $1491625364964\cdots$ What is the $612^{th}$ digit?

Let $A$ and $B$ be two positive integers and $A=B^2$. If $A$ satisfies the following conditions, find the value of $B$:

• $A$'s thousands digit is $4$
• $A$'s tens digit is $9$
• The sum of all $A$'s digits is $19$

Is it possible to find four positive integers such that $2002$ plus the product of any two of them is always a square? If yes, find such four positive integers. If no, explain.

If the middle term of three consecutive integers is a perfect square, then the product of these three numbers is called a $\textit{beautiful}$ number. What is the greatest common divisor of all the $\textit{beautiful}$ numbers?

Find the smallest square whose last three digits are the same but not equal $0$.

Let $\overline{ABCA}$ be a four-digit number. If $\overline{AB}$ is a prime, $\overline{BC}$ is a square, and $\overline{CA}$ is the product of a prime and a greater-than-one square. Find all such $\overline{ABCA}$.

Let $A$ be a two-digit number, multiplying $A$ by 6 yields a three-digit number $B$. The difference of the two five-digit numbers obtained by appending $A$ to the left and right of $B$, respectively, is a perfect square. Find the sum of all such possible $A$s.

Find such a positive integer $n$ such that both $(n-100)$ and $(n-63)$ are square numbers.

Find such a positive integer $n$ such that both $(n+23)$ and $(n-30)$ are square numbers.

Find the smallest positive integer $n$ such that $\frac{12!}{n}$ is a square.

Consider the following $32$ numbers: $1!, 2!, 3!, \cdots, 32!$. If one of them is removed, then the product of the remaining $31$ numbers is a perfect squre. What is that removed number?

There exist $5$ consecutive positive integers such that their sum is a square, and the sum of the middle three is a cube. What is the smallest one of these five numbers?

Find a $4$-digit square number $x$ such that if every digit of $x$ is increased by 1, the new number is still a perfect square.