Need some math confirmation from a math whiz - 3 series of numbers

[NewJeffCT]

Series A is 1, 2, 3, 4... 28
Series B is 1, 2, 3, 4... 34
Series C is 1, 2, 3, 4... 80

If each series starts at 1 and continually repeats itself, how many times through Series A will it be until all three series' are at "1" again at the same time? (meaning, Series A resets back to "1" when Series B and C are at "29", and then Series B resets to "1" when Series C is at "35" and Series A is now on "6&quot

As a second challenge, when does each series' halfway point coincide? Meaning, when is Series A at "14" while Series B is at "17" and Series C is at "40" ?

For the first problem, I came up with 340 times through Series A, with "1" coming up at the beginning of the 341st time through Series A. Is there an easy way to calculate? (I did it by copying & pasting in Excel and some IF statements)

I haven't calculated the second part yet.

Thanks a lot

[Baitball Blogger]

1. Major brain melt.

Let me know when I can highlight the white for the answer.

[NewJeffCT]

2. No white to highlight

I'm trying to come up with the answer myself.

[ret5hd]

3. I believe it will have something to do with the lowest common multiple of 28, 34, and 80.

For the halfway point, maybe some fraction or root of the lowest common multiple.

[NewJeffCT]

6. yes

if you take 14, 17 and 40 and multiply them, you get 9,520. 9,520/28 is 340

[pokerfan]

4. I also get 340

by taking the LCM of 28, 34 and 80, then dividing by 28.

For the second problem, I would probably start with taking the LCM of 14, 34 and 40, then discarding the solutions that have even number of cycles as only the odd ones would represent halfway points.

[NewJeffCT]

5. Thanks

I'll see what I can come up with for the second part as well.

[struggle4progress]

7. For the first problem, ask when 28*a = 34*b = 80*c

Solutions occur whenever a = 17*20*n = 340*n, b = 14*20*n = 280*n, c = 7*17*n = 119*n due to the fact that
28*17*20 = 14*34*20 = 7*17*80

For the second part, I think you may need to find an odd number that is even

[NewJeffCT]

8. Thanks

And, for the second part, I figured if you went through the series enough times, you might be able to have them at the halfway point together at some time.

[struggle4progress]

9. For the second problem, ask yourself when 28*a + 14 = 34*b + 17 = 80*c + 40

which corresponds to passing "a and a half times" thru the first series, "b and a half times" thru the second series, and "c and a half times" thru the third series

For more info, look up the "Chinese Remainder Theorem"