Calculating M and the Unit Digit of M^2003

[anemone]

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]

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

 

[kaliprasad]

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]

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

Hi kaliprasad,

Thanks for taking the time to participate in this challenge problem and I can tell how much you enjoyed working with some of the problems that I posted here and in case if you have any interesting mathematics problems to share with us, please feel free to do so! :p

Just a moment and I'll have it. I just have to program Excel...

-Dan

Hi Dan,

Thank you for the reply and you know what, you're one of the clever \(\displaystyle \cap\) humorous member at MHB!

 

[CellCoree]

The expression given for M is a complex one, involving both real and absolute value operations. To accurately calculate M, we would need to know the value of x. Without this information, it is not possible to provide a specific numerical value for M.

However, we can make some general observations about the behavior of M. First, we can see that the denominator of the first fraction, 1+x, will always be positive since x is a real number. This means that the overall value of the fraction will be affected by the sign of the numerator, 3x-1. Similarly, the denominator of the second fraction, |2-x|, will also always be positive, so the overall value of the second fraction will be affected by the signs of the two square root terms.

Next, we can consider the behavior of the absolute value terms. For |x| to be greater than 2, x must be either greater than 2 or less than -2. In either case, the square root terms in the second fraction will be complex numbers. This means that the overall value of the second fraction will be a complex number, and the overall value of M will also be a complex number.

As for the unit digit of M^{2003}, we can use the properties of modular arithmetic to determine that the unit digit of M^{2003} will be the same as the unit digit of M^3. This is because the unit digit of any power is determined by the unit digit of the base raised to that power. So, to find the unit digit of M^{2003}, we would need to know the unit digit of M^3.

Without knowing the value of x, we cannot determine the exact value of M^3 or the unit digit of M^{2003}. However, we can make some general observations about the unit digit of M^3. Since M^3 is a complex number, it is likely that the unit digit will also be complex. Additionally, the unit digit of M^3 will be affected by the signs of the numerator and square root terms, similar to M.

In conclusion, without knowing the value of x, we cannot provide a specific numerical value for M or the unit digit of M^{2003}. However, we can make some general observations about their behavior and determine that they are likely to be complex numbers. Further analysis and calculations would be needed to determine the exact values.