5. Find the greatest common divisor and the least common multiple of the following expressions:

$$
4^{x}-9^{x}, \quad 4^{x}+2 \cdot 6^{x}+9^{x}, \quad 4^{x}+3 \cdot 6^{x}+2 \cdot 9^{x}, \quad 8^{x}+27^{x}
$$

Solve each problem independently on the attached paper, and each problem on a separate page.

Do not sign the sheets, write only your code.

Write your answers with a pen clearly and neatly. Each problem will be graded on a scale from 0 to 6 points.

4th National Mathematics Competition for Students of Secondary Technical and Vocational Schools

Maribor, April 17, 2004

## PROBLEMS FOR 2nd GRADE
5. First, we factorize each expression: $4^{x}-9^{x}=\left(2^{x}+3^{x}\right)\left(2^{x}-3^{x}\right)$, $4^{x}+2 \cdot 6^{x}+9^{x}=\left(2^{x}+3^{x}\right)^{2}$, $4^{x}+3 \cdot 6^{x}+2 \cdot 9^{x}=\left(2^{x}+3^{x}\right)\left(2^{x}+2 \cdot 3^{x}\right)$, and $8^{x}+27^{x}=\left(2^{x}+3^{x}\right)\left(4^{x}-6^{x}+9^{x}\right)$. We see that the greatest common divisor is $2^{x}+3^{x}$, and the least common multiple is $\left(2^{x}+3^{x}\right)^{2}\left(2^{x}-3^{x}\right)\left(2^{x}+2 \cdot 3^{x}\right)\left(4^{x}-6^{x}+9^{x}\right)$.

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-07.jpg?height=60&width=1636&top_left_y=2534&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-07.jpg?height=63&width=1636&top_left_y=2581&top_left_x=210)

Factorization: $4^{x}+3 \cdot 6^{x}+2 \cdot 9^{x}=\left(2^{x}+3^{x}\right)\left(2^{x}+2 \cdot 3^{x}\right) \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-08.jpg?height=83&width=1639&top_left_y=227&top_left_x=206)

Correct notation for $D$ and $v$

## Second Year