Discussion Overview
The discussion revolves around calculating the value of $$M$$ defined by a specific expression involving a real number $$x$$, and determining the unit digit of $$M^{2003}$$. The scope includes mathematical reasoning and problem-solving related to algebra and modular arithmetic.
Discussion Character
- Mathematical reasoning
- Exploratory
- Homework-related
Main Points Raised
- Participants define $$M$$ as $$M=\frac{3x-1}{1+x}-\frac{\sqrt{\mid x\mid-2}+\sqrt{2-\mid x \mid}}{\mid 2-x \mid}$$ and seek to find its value and the unit digit of $$M^{2003}$$.
- One participant argues that since the square root of $$|x| - 2$$ must be zero, it leads to the conclusion that $$|x| - 2 = 0$$, resulting in potential values of $$x = 2$$ or $$x = -2$$.
- It is noted that $$x$$ cannot be 2 due to the denominator $$|2-x|$$, leading to the conclusion that $$x = -2$$ is the only viable option.
- Substituting $$x = -2$$ into the expression for $$M$$ yields $$M = 7$$.
- Further calculations show that $$M^4 \equiv 1 \mod 10$$, leading to $$M^{2000} \equiv 1 \mod 10$$ and thus $$M^{2003} \equiv M^3 \mod 10$$, which is calculated as $$343 \mod 10$$, resulting in a unit digit of 3.
- Another participant expresses appreciation for the problem and invites further mathematical contributions from others.
Areas of Agreement / Disagreement
There appears to be a consensus on the calculation of $$M$$ when $$x = -2$$, leading to the unit digit of $$M^{2003}$$ being 3. However, the discussion does not explore alternative values for $$x$$ beyond those mentioned.
Contextual Notes
The discussion assumes certain properties of the square root and absolute value functions without delving into potential edge cases or alternative interpretations of the expression for $$M$$.
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