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Find the smallest number that is made up of each of the digits 1 through 9 exactly once and is divisible by 99.

 Dec 29, 2019
 #1
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A nunber is divisible by 99 if it is divisible by 9 and 11.

 

Divisibility rule for 9 is if all the numbers sum up to a multiple of 9, then the number is divisible by 9.

 

Any number made of digits 1 through 9 is divisilble by 9.

 

 

so basically the question is now:

Find the smallest number using digits 1-9 that is divisible by 11.

 

We start with 123456789. Because its the smallest.

 

Using the picture above, we get

 

1 + 3 + 5 + 7 + 9 = 25

 

2 + 4 + 6 + 8 = 20

 

25 - 20 = 5. Which is not divisible by 11.

 

So how can we manipulate the red and blue numbers to create a result that is divisble by 11?

 

Welp, guess and check...

Sorry, that is the best I can do. I was able to at least limit the number of guesses  at least....

 Dec 30, 2019
 #2
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The smallest number I could find is:

 

123,475,869 mod 99 = 0

 Dec 30, 2019

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