# Find M

#### anemone

##### MHB POTW Director
Staff member
Let $$\displaystyle x$$ be a real number and let $$\displaystyle M=\frac{3x-1}{1+x}-\frac{\sqrt{\mid x\mid-2}+\sqrt{2-\mid x \mid}}{\mid 2-x \mid}$$.

Find $$\displaystyle M$$ and also the unit digit of $$\displaystyle M^{2003}$$.

#### topsquark

##### Well-known member
MHB Math Helper
Let $$\displaystyle x$$ be a real number and let $$\displaystyle M=\frac{3x-1}{1+x}-\frac{\sqrt{\mid x\mid-2}+\sqrt{2-\mid x \mid}}{\mid 2-x \mid}$$.

Find $$\displaystyle M$$ and also the unit digit of $$\displaystyle M^{2003}$$.
Just a moment and I'll have it. I just have to program Excel...

-Dan

##### Well-known member
Let $$\displaystyle x$$ be a real number and let $$\displaystyle M=\frac{3x-1}{1+x}-\frac{\sqrt{\mid x\mid-2}+\sqrt{2-\mid x \mid}}{\mid 2-x \mid}$$.

Find $$\displaystyle M$$ and also the unit digit of $$\displaystyle M^{2003}$$.
We are taking square root of |x| - 2 and its –ve so it has to be zero
So |x| - 2 = 0 or x = 2 or – 2
x cannot be 2 as |2-x| is in denominator
so x = - 2
hence putting the value x = -2 we get M = 7
as M^4 = 1 mod 10
M^2000 = 1 mod 10 or M^2003 = M^3 mod 10 = 343 mod 10 or 3
3 is the unit digit

#### anemone

##### MHB POTW Director
Staff member
We are taking square root of |x| - 2 and its –ve so it has to be zero
So |x| - 2 = 0 or x = 2 or – 2
x cannot be 2 as |2-x| is in denominator
so x = - 2
hence putting the value x = -2 we get M = 7
as M^4 = 1 mod 10
M^2000 = 1 mod 10 or M^2003 = M^3 mod 10 = 343 mod 10 or 3
3 is the unit digit
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