What is the smallest positive integer whose reciprocal contains all 10 digits in the ten consecutive digits starting with the first nonzero digit?
a) smallest when truncated (whether rounding gives same result or not)
b) smallest when rounded (whether truncation gives same result or not)
c) smallest when truncation and rounding give the same result
Note: some algorithms round numbers with a fractional part of exactly 0.5 to the nearest even integer, but for this exercise, 0.5 is always rounded up.
The first fifteen that qualify using truncation are:
n reciprocal tested sequence
648 0.001543209876543 1543209876
6480 0.000154320987654 1543209876
6532 0.000153092467851 1530924678
17229 0.000058041673922 5804167392
25235 0.000039627501486 3962750148
26847 0.000037248109658 3724810965
31264 0.000031985670420 3198567042
32712 0.000030569821472 3056982147
37257 0.000026840593714 2684059371
38250 0.000026143790850 2614379085
40502 0.000024690138759 2469013875
42626 0.000023459860179 2345986017
43177 0.000023160478959 2316047895
45894 0.000021789340655 2178934065
50625 0.000019753086420 1975308642
The first fifteen that qualify using rounding are:
1608 0.000621890547264 6218905473
1637 0.000610873549175 6108735492
2025 0.000493827160494 4938271605
3825 0.000261437908497 2614379085
10125 0.000098765432099 9876543210
11722 0.000085309674117 8530967412
16080 0.000062189054726 6218905473
16370 0.000061087354918 6108735492
17229 0.000058041673922 5804167392
20250 0.000049382716049 4938271605
20717 0.000048269537095 4826953710
28188 0.000035476089116 3547608912
28834 0.000034681279046 3468127905
31264 0.000031985670420 3198567042
31492 0.000031754096278 3175409628
17229 is the smallest integer that qualifies both ways, as its reciprocal, 0.000058041673922 has the digits '5804167392' either way as the next digit is only 2 and does not cause the rounding to affect what comes before.
clearvars,clc
ct=0;
for n=5:90000
r=1/n;
s=strrep(sprintf('%17.15f',r), '0.', '');
tst= num2str(str2num(s)) ;
if length(tst)>9
t=tst(1:10);
if length(unique(t))==10
fprintf('%7d %17.15f %s\n',n, r, t)
ct=ct+1;
if ct==15
break
end
end
end
end
clearvars
disp(' ')
ct=0;
for n=5:90000
r=1/n;
s=strrep(sprintf('%17.15f',r), '0.', '');
tst= num2str(str2num(s)) ;
if length(tst)>9
t=num2str(str2num(tst(1:11))+5);
t=t(1:10);
if length(unique(t))==10
fprintf('%7d %17.15f %s\n',n, r, t)
ct=ct+1;
if ct==15
break
end
end
end
end
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Posted by Charlie
on 2025-04-16 14:40:10 |
computer solution
