Alan

 lcm(8*512, 10^10) = 40 000 000 000

 lcm(512, 10^10) = 10 000 000 000

I think 8m is meant to be interpreted as 8*m, not 8000+m.

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"If m is a 3-digit positive integer such that , then what is the value of m?"

512

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Use the fact that a^2 - b^2 = (a+b)(a-b) so we have:  (2001+1992)*9/[(2003+1997)*6]

or 3993*9/(4000*6) ≈ 1.497

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In general:  \(e^{i\theta}=\cos \theta+i\sin\theta\)

You already know that  \(\sqrt{i}=e^{i\pi /4}\)

Put these together to get:  \(\sqrt{i}=\cos{(\pi /4)+i\sin{(\pi /4)}}\)

Since we have: \(\cos{(\pi /4)}=\frac{1}{\sqrt{2}} \space \text{and }\sin{(\pi /4)}=\frac{1}{\sqrt{2}}\)

we can see that:  \(\sqrt{i}=\frac{1+i}{\sqrt{2}}\)

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Here's a picture to illustrate Melody's answer:

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The n'th term is given by an = a1 + (n-1)d

Here:   

9 = a1 + 4*d

-84 = a1 + 31*d

These allow you to find a1 and d, so you can now find: 

a23 = a1 + 22*d

To get the current value multiply $72000 by 1.35

If the coordinates of u and v represent the heads of vectors that start at the origin, then the slope of u = 7/4 and the slope of v = -8/28 → -2/7

The slopes are not the same hence the lines are not parallel.

The slope of v is not -1/the slope of u, hence the lines are not orthogonal.

$(2900*1600 + 4200*1100)*1.035 → $9 584 100