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What is the largest positive integer with only even digits that is less than 10,000 and is a multiple of 9?

thank you!

 Nov 6, 2019
 #1
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There are 24=16 four-digit numbers containing only the digits 1 and 2. The same for all other of the (92)=9⋅82=36 combinations of 2 of the 9 possible digits. If you add all those together, you will have counted the number "1111" and "2222" and so on too often. The highest possible integer in this context is 8,820.

 Nov 6, 2019
 #2
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Here's another  (less neat ) solution

 

We can guess that   the answer   might be    88 _ _

 

Note that   8901  is a multiple of 9   because the sum of its digits is divisible by 9

 

We need to subtract  some odd multiple of 9 from this to produce an integer with all even digits

 

The possibilities are 9, 27, 45, 63, 81

 

81  is the one we need

 

So

8901  - 81  =

 

8820  !!!   (as the guest found)

 

 

cool cool cool

 Nov 6, 2019
edited by CPhill  Nov 6, 2019

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