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Integers and their sums of digits (Posted on 2017-03-13) Difficulty: 2 of 5
Find all solutions of the system
s(x) + s(y) = x
x + y + s(z) = z
s(x) + s(y) + s(z) = y − 4,
where x, y and z are positive integers, and each of s(x), s(y) and s(z) denotes the sum of digits of x, y and z, respectively, in decimal notation.

No Solution Yet Submitted by Ady TZIDON    
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Hints/Tips Analytical Approach Comment 3 of 3 |
My solution, done before Ady's hint, matches Charlie's.
I just have not written it up yet.

I observed that y must be a multiple of 9 (from the 1st equation).

(Also, then, x must be a multiple of 9 from the second equation, although this inference was not used).

Substituting equations 1 and 2 into 3 yields
z = 2y - 4

From equation 2, x = z - s(z) - y 

So, it is easy to build a table of z and x for y = 9, 18, 27, 36, etc.
x and y cannot be bigger than 100, so the table is qyuite manageable.

Testing those entries using equation 1, yields just two solutions

  Posted by Steve Herman on 2017-03-19 11:56:47
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