Find three positive integers with the following two properties:
- The sum of any two of them has digit sum less than 15
- The sum of all three integers has digit sum more than 200
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$\begingroup$ By sum do you mean multiplication? $\endgroup$– Prince DeepthinkerAug 19, 2020 at 14:40
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$\begingroup$ Exclusively that is $\endgroup$– Prince DeepthinkerAug 19, 2020 at 14:41
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2 Answers
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Here is a possible answer which I think suggests the general strategy
4554554554554554554554554554554554554555
5455455455455455455455455455455455455455
5545545545545545545545545545545545545545
The pairwise sums are
10010010010010010010010010010010010010010
10100100100100100100100100100100100100100
11001001001001001001001001001001001001000 (all digit sums are 14)
While the overall sum is
15555555555555555555555555555555555555555 (digit sum 201)
answered Aug 19, 2020 at 14:59
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Take these three numbers:
$$a = 4444444444444\ 5555555555555\ 5555555555555\ 5\\b = 5555555555555\ 4444444444444\ 5555555555555\ 5\\c = 5555555555555\ 5555555555555\ 4444444444444\ 5$$ They all consist of three blocks of 13 digits, of which one is all fours and the rest all fives, followed by an extra five.
The pairwise sums are:
$$a+b = 1\ 0000000000000\ 0000000000000\ 1111111111111\ 0\\b+c = 1\ 1111111111111\ 0000000000000\ 0000000000000\ 0\\c+a = 1\ 0000000000000\ 1111111111111\ 0000000000000\ 0$$ The sum of any two of them gives a number with a leading $1$ and block of thirteen $1$s, for a digit sum of $14$.
The sum of all three is:
$$a+b+c = 1\ 5555555555555\ 5555555555555\ 5555555555555\ 5$$ which is a one followed by $40$ fives, for a digit sum of $201$.
answered Aug 19, 2020 at 15:03