Math Contest #13 Results and Solution

Solution

The problem of this contest was a bit harder and I'm not overly surprised that nobody completely solved every task. I'll give here a short summary how you could have solved them:

1. This can be done with the 1-digit sum:
   123456789 → 1+2+3+4+5+6+7+8+9 = 45, 45 is divisible by 3 → yes
   4327384926 → 48 → yes
   7777777777 → 10*7 = 70 → no
2. For this you needed to apply the new concept of 2-digit sums:
   308 → 3+08 = 11 → yes
   453825 → 45+38+25 = 108 → no
   187869 → 18+78+69 = 165 → yes
3. This was a bit more tricky, because you would need to either guess or find the concept asked in 5. to figure out which digit sum to use.
   909*11 = 9999 = 10⁴ → you needed to use the 4-digit sum:
   46385600 → 4638+5600 = 10238 Here it is not so obvious, so I repeat the digit sum: 1+0238 = 239 → no
   36369999 → 3636+9999 = 13635 I repeat it again: 1+3635 = 3636 = 4*909 → yes
   4072320 → 407+2320 = 2727 = 3*909 → yes
4. a)
   I thought this task would be the most easy.
   You wouldn't even need to know anything about digit sums to solve this :P.
   If you know a number is divisible by two factors which have no common prime factor, you also know that the number is also divisible by the product of those 2 factors:
   3*11 = 33 and 9*11 = 99
   Easy, right?
   b)
   This one could again only be solved with the knowledge of five.
   You can check for all factors of 10³ - 1 = 999 with the 3-digit sum. Those factors are:
   3, 9, 27, 37, 111, 333, 999
5. With any n-digit sum you can check for all factors of 10^n -1
   Why is that?
   You might need to figure it out yourself in a future contest. Until then I will keep it.

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What is your chance of winning:

p(You Win) = 1/n, n = number of entries

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List of participants with their entries:

Name	solutions found	comment
@tonimontana	2. 5.	10 STEM for you!
@mmunited	1. 2. 3.	Although you didn't get the concept of 3., you got the right results → 7 STEM for you!
@golddeck	1. 2. (4.)	You got many errors in 4., but you also got a few correct things. So I'll give 1.25 STEM for that task. → 4.25 STEM for you!

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Winner draw:

Congratulations @golddeck, you won 1 SBI!
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The next contest starts tomorrow. Don't miss it!