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NIH New user 11 Posts |
Walking through the woods one day you notice a scrap of paper on the ground, on which is written "37! = 13763753091226345046315979581########0000000", where each "#" indicates a digit which is smudged beyond recognition.
Your curiosity aroused, you decide to calculate the missing digits. Unfortunately, you have only a pen and a sheet of paper at hand. How do you proceed, assuming you wish to minimise arithmetical effort? (Assume that the digits given are correct.) |
Scott Cram Inner circle 2678 Posts |
First, we need to figure out how many zeroes 37! (37 factorial, or 37 * 36 * 35 * . . . * 2 * 1) has. Using the standard process, we can take into account that there are 8 zeroes at the end of 37 factorial. (I can explain this if you like, but it will take a while to explain).
There are seven zeroes readable on the paper, so there must be one more zero as the rightmost smudged number. This gives us this number (the new numbers will be in italics): 13763753091226345046315979581#######00000000 Next, we know that 37 factorial will be a multiple of 9 (because when you make a factorial of any whole number greater than 6, it will by definition be a multiple of 9), so it's digital root will be 9. The numbers showing have a digital root of 8, so the remaining 7 digits must have a digital root of 1. We also know that the first digit to the left of the eight zeroes can't be a zero itself, because otherwise the number would end in nine zeroes, not eight. The other 6 mystery digits can be zeroes, however. I'm getting stuck here, so I'll have to think about this some more. |
NIH New user 11 Posts |
Divisibility tests are key to this puzzle, but we need to use more than just divisibility by 9 (digital roots.)
As a smaller example, given that 11! = 39ab6800, find the missing digits a and b. Note that 11! is divisible by both 9 and 11, and hence by 99. So we can use the base 100 equivalent of base 10 digital roots. That is, the sum of digit pairs must be divisible by 99. Hence 00 + 68 + (10 + b) + 39 is divisible by 99. It follows that 10a + b = 91, and that a = 9, b = 1. |
NIH New user 11 Posts |
I added this puzzle, and the solution, to my website: http://www.qbyte.org/puzzles/puzzle09.html#p84
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