Given that for a positive integer N, the two relationships:
sod(N) = 100, and sod(44*N) = 800
are satisfied simultaneously.
Determine sod(3*N), providing valid reasoning for your answer.
Notes:
i) sod(m) denotes the sum of the digits of m, where m is a positive integer.
ii) Computer program and spreadsheet assisted methodologies are welcome, but an analytical solution is preferred.
iii) Adapted from a problem which appeared at a Russian Mathematical Olympiad.
Constraint: sod(44*N) = 8*sod(N)
In general, for the constraint to be true, N must be of the form x0x0x0x0x0x0x where any x can be either 1 or 2, the final x is optional, and any 0 can be replaced by multiple zeros.
The smallest such number is 99 digits long: 202020...02 where there are 50 2s alternating with 49 zeros. Then 44N is a string of 100 8s.
The largest n satisfying the condition,
for a given number of digits, is of the form 20202020202; the fractional part of the base 10 log of this is .30539480106643.
And the base 10 log of 44 is about 1.6434526764861872. Since the sum of those 2 logs is a little less than 2, 44N will always be 1 digit longer than N (never 2 digits longer)
All such numbers appear to be formed solely of digits {0,1,2}; if such a number is multiplied by 3, the sod is also multiplied by 3 since there is never a carry digit. So if the sod(N) is 100, then sod(3N) = 300
I did check all integers from 1 to 50,000,000 if sod(44n) equals 8*sod(n), then in every case, sod(3n) equals 3*sod(n).
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Posted by Larry
on 2023-04-26 11:52:20 |
No Subject
