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Feisty Fifty-six (Posted on 2013-10-04) Difficulty: 3 of 5
Determine the minimum value of a positive integer N such that:
N ends with 56, and:
N is divisible by 56 and:
The sum of the digits of N is equal to 56.

Extra Challenge: Solving this puzzle without a computer program.

No Solution Yet Submitted by K Sengupta    
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Solution By hand method | Comment 4 of 5 |
We know the number looks like _ _ ..._ _ 56 and the sum of the digits besides the last two is 45.
Since 56=7*8 we need the number to be divisible by 8.  The third to the last digit must then be even.
To make the smallest digits at the front try to make this third to last digit be 8:

_ _ ... _ 856
The remaining digits must sum to 37.  This can't be done with 4 digits so the best we can do is
_ _ _ _ _ 856
The blanks must now also form a 5-digit number that gives a remainder of 2 when divided by 7.
If you try to start this number with a 1 you can only get 19999 which doesn't work (0 mod 7)
So maybe it can start with 2.
2 _ _ _ _
where the remaining digits sum to 35.  three 9s and an 8.
28999 no
29899 yes

The number sought is

29899856

  Posted by Jer on 2013-10-04 16:11:41
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