- #1
kaliprasad
Gold Member
MHB
- 1,335
- 0
Find all integers x such that $x(x+1)(x+7)(x+8)$ is square of an integer
my solution:kaliprasad said:Find all integers x such that $x(x+1)(x+7)(x+8)$ is square of an integer
Albert said:my solution:
$x(x+1)(x+7)(x+8)=144---(5)$
kaliprasad said:How ?
Albert said:from (2)
$(x+1)(x+8)=x(x+7)$
$x=-4,(x+1)(x+8)=x(x+7)=-12$
$\therefore x(x+1)(x+7)(x+8)=(-12)^2=144$
[sp]Let $y = x+4$. Then $$\begin{aligned} x(x+1)(x+7)(x+8) &= (y-4)(y+4)(y-3)(y+3) \\ &= (y^2-16)(y^2-9) \\ &= y^4 - 25y^2 + 144 \\ &= (y^2-12)^2 - y^2. \end{aligned}$$ If that is a square then either $y=0$ or $y^2$ must be at least as large as the difference between $(y^2-12)^2$ and the previous square $(y^2-13)^2$. But $(y^2-12)^2 - (y^2-13)^2 = 2y^2 - 25.$ So we must have $y^2 \geqslant 2y^2 - 25$, which means that $y^2 \leqslant 25.$ Therefore $y$ lies between $-5$ and $5$, so that $x$ lies between $-9$ and $1$. The valid solutions in that interval are exactly those found by Albert.[/sp]kaliprasad said:Find all integers x such that $x(x+1)(x+7)(x+8)$ is square of an integer
A square integer is a number that can be expressed as the product of two equal integers. For example, 9 is a square integer because it can be written as 3x3. This also means that the number has an exact square root.
To determine if x(x+1)(x+7)(x+8) is a square integer, you would need to simplify the expression and see if it can be written as the product of two equal integers. If it can be, then it is a square integer.
No, not all values of x will result in a square integer. For example, if x=2, the expression would simplify to 168, which is not a square integer. However, there are certain values of x that will result in a square integer, such as x=0 or x=-1.
Yes, there are various methods for determining if an expression is a square integer. One method is to factor the expression and see if it can be written as the product of two equal integers. Another method is to take the square root of the expression and see if it results in a whole number.
Knowing if an expression is a square integer can be useful in various scientific fields, such as in number theory or in solving certain mathematical equations. It can also be helpful in understanding patterns and relationships between numbers. Additionally, the concept of square integers is often used in physics and engineering, particularly in calculations involving areas and volumes.