Math Olympiad problem (easy?)

[sirio11]

From my own creation to those interested in Olympiad problems in this forum;

Let
S(1)=1+2+3+......+2007
S(2)=1^2+2^2+3^2+......+2007^2

S(n)=1^n+2^n+3^n+......+2007^n

Let A(n) to be equal to the last digit of the number S(n) and let

D = A(1)+A(2)+A(3)+.....+A(2007)

What's the last digit of the number D?

[Phil153]

I love these, havent done them in ages. This one was fairly easy. Answer in white:

<font color="white">zero.

The key is in noticing a number of repeating elements:

- The last digit of the nth power will form a sequence that repeats on each ten. (i.e 2^36 has the same last digit as 12^36)
- The last digit of the sum of the nth power repeats for every four values of n. You can see this by writing out the 10 digits.

1st power: 1234567890 Last digit of sum to 2007: 8 -|
2nd power: 1496569410 Last digit of sum to 2007: 0 -|} last digit of sum = 8
3rd power: 1874563290 Last digit of sum to 2007: 4 -|}
4th power: 1616561610 Last digit of sum to 2007: 6 -|
5th power: 1234567890 Last digit of sum to 2007: 8
.
.
etc

So it's a simple matter of dividing 2007 by 4, which gets us to 2004 with a last digit of 8, and 2005,2006 &amp; 2007 remaining. These sum to (8+0+4) = 12, so the last digit is 8+2 = 0.

If I didn't screw up the arithmetic.
</font>

[Drag]

The final sum can be regrouped in the rows
T(1)=1+1^2+1^3+...1^2007
T(2)=2+2^2+2^3+...2^2007
...
T(2007)=2007+2007^2+2007^3+...2007^2007

S=T(1)+...+T(2007)

As we are interested only in the final digit we can replace
evrywhere T(10+x)=T(x)for x&gt;0. Repeating this process we get the sequence:
T(1), T(2), ..., T(9), T(1), T(2), ... , T(9), ...

All the T(x) that get repeated 10 times can be deleted.

So the original problem is equivalent to the problem of finding the last digit of the following sum:
Dx= T(1) + T(2) + ... + T(7)

This problem is quite trivial.

P.S. In this reduction we used the fact that operation of taking the last digit is linear in respect to the summation.

Edit:
To put the final answer, I also got 0.

[thylacine]

Why do you call this a "Math Olympiad problem"?