Day 24: The Missing Digit
Rudolph was tired of pulling the sleigh everywhere.
So, using his awesome intellect, he calculated 2 0 1 5 ! and then wrote his answer down on a piece of paper. But unfortunately he missed out one digit and then couldn't remember what it was.
His (also intelligent) friend tried to help him and found that the sum of the digits was 23514.
What is the missing digit?
This problem is part of the Advent Calendar 2015 .
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1 solution
How did you came upon the thinking that 2015! Is divisible by 9
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Good question! It's because 2 0 1 5 ! = 2 0 1 5 × 2 0 1 4 × ⋯ × 1 0 × 9 × 8 × ⋯ × 2 × 1
9 is part of the product, so 2 0 1 5 is divisible by 9 .
We can extend this to show the useful, but often overlooked, fact that n ! is divisible by all integers from 1 to n .
2015!=1 2 3 ..... 7 8 9 10 11 12....... 2014*2015
Same way. Nice question.
Really enjoyed this question! I used digital roots instead.
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Ah yes, thank you! Digital roots are very useful indeed.
Yes Same Way, btw 23514 is any random number right? :p
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Ah no, I had to make sure that the problem was correct. The sum of the digits really is 2 3 5 1 7 ; in some coincidence it's quite nice that 2 3 5 1 4 contains all the integers from 1 to 5 which is really cool :)
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How did you Find the digit Sum?
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@Kushagra Sahni – I used Wolfram Alpha for a quick answer (although I could have used Python) :)
did same..
What if we consider 2015! is divisible by 3 (and sum of the given digits 15 which is divisible by three). So, why can't the missing number be zero?
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That's because if you only consider divisibility by 3, it could be 0, 3, 6 or 9. So you must look for something that narrows the choices down. So now consider divisibility by 9 and you will get the correct answer. Hope this helps!
2 0 1 5 ! is divisible by 9 , so the sum of its digits must be too.
The sum of the digits of 2 3 5 1 4 is 1 5 , so the missing digit must be 3 to make the sum of the digits divisible by 9 ( 1 5 + 3 = 1 8 ), and to make 2 0 1 5 ! divisible by 9 too.
So the answer must be 3 .