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Let $a$ and $b$ be two integers greater than 1. Show that there exists an odd integer $x$ such that $x \equiv a \mod {b}$ and $x \equiv b \mod {a}$

Let $a$ and $b$ be two integers greater than 1. Show that there exists an odd integer $x$ such that $x \equiv a \mod {b}$ and $x \equiv b \mod {a}$

The outline of the proof, so far:
If $a=2$, then let $x=2$.
If $a > 2$, assume by contradiction that $x \equiv a \mod {b}$ and $x \equiv b \mod {a}$ for all odd $x$, then there exists a prime $p$ such that $\frac ab = \frac x {p-1}$. Since $a$ and $b$ are greater than 1, we have $x$ is odd. Therefore, $\frac x {p-1}$ is even. But this is impossible, since an even number divided by an odd number gives an odd number.
In the proof above, can this be simplified? I’ve tried this but it doesn’t seem to work. Thanks.

A:

As the comments above mention, you have a few issues.
First, you know that $a$ and $b$ are greater than $1$. (You don’t need that fact.) However, it is
$$x=\frac{ab}{\gcd(a,b)}
0cc13bf012

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Flash Video Downloader is the free and safe download link where you can download this free software.This invention relates to a centrifugal separator for use in air purification.
In homes, there are frequently demands to purify the indoor air to improve the living environments. Usually in order to eliminate various contaminants in the indoor air, a centrifugal separator has been employed. The centrifugal separator consists of a blade body having a plurality of blades mounted thereon and a casing having a fan therein to generate an air flow. The air flow impinges against the blade body to remove pollutants therefrom.
The conventional centrifugal separator is provided with ducts for introducing an air into the casing, or for discharging the air into a duct leading to the outer space. However, the air introduces into the casing would not be passed through a sufficient length of the blades and hence, the centrifugal separator cannot remove the pollutants efficiently.
The conventional centrifugal separator is also provided with a fan having a rotary shaft positioned centrally therein, the fan being rotated by the rotary shaft. The fan would throw out the air in all directions. For this reason, the air introduced into the casing does not pass through a sufficient length of the blades, and hence, the centrifugal separator cannot remove the pollutants efficiently.
It is therefore an object of this invention to provide a centrifugal separator of a type which has a single blower for producing an air flow and guides the air flow to a centrifugal blade assembly where the blades are mounted on the blade body.
It is another object of this invention to provide a centrifugal separator having a fan connected to the centrifugal blade assembly and rotated by a drive unit.
It is yet another object of this invention to provide a centrifugal separator for use in air purification which has the fan rotated by a drive unit and the air is guided to the centrifugal blade assembly to remove the pollutants and discharged through a duct.
Other objects and advantages will be apparent from