n is a positive integer base from 2 to 36 inclusively.
Each of α and β is a
nonzero digit of base n from 1 to n-1 inclusively.
Determine all possible triplets (α, β, n) such that:
(α*β)base n
----------- is a positive integer.
(α.β)base n
### For example, in base ten corresponding to (α, β) = (4, 5), we observe that:
(4*5)/(4.5) = 40/9, which is NOT a positive integer.
(In reply to
re: computer solution - 3 other solutions by Larry)
I hadn't realized that declaring a, b, f, numerator, and denominator as syms was not sufficient. I had to make sure syms were placed in them; there's no conversion taking place. Only when operations are performed with ordinary numbers does conversion to sym take place for the result. Some numbers were small fractions off.
Python, which you use, uses indefinitely long integers by default.
Total of 77 appears at the bottom.
New version:
clearvars,clc
ct=0;
syms n a b numerator denominator f
for n=2:36
for a=1:n-1
for b=1:n-1
numerator=sym(a)*b;
denominator=sym(a)+sym(b)/n;
f=numerator/denominator;
if round(f)==f
% disp([a b n numerator denominator f])
sout=sprintf('%2d %2d %2d %13.9f %13.9f %13.9f \n',a, b, n, numerator, denominator, f);
trans=false; ppos=0;
for i=1:length(sout)
switch sout(i)
case '.'
trans=true; ppos=i;
case {' ' '\n' }
if trans==true
trans=false;
for j=i-1:-1:ppos
if sout(j)>'0' && sout(j) <='9'
break
end
if sout(j)=='0'
sout(j)=' ';
else
if sout(j)=='.'
sout(j)=' ';
break
end
end
end
ppos=0;
end
end
end
fprintf('%s',sout);
ct=ct+1;
end
end
end
end
disp(ct)
for n=2:36
for a=1:n-1
for b=1:n-1
if n==15 && a==1 && b==10
n=n;
end
numerator=sym(a)*b;
denominator=a+sym(b)/n;
f=numerator/denominator;
if round(f)==f
ab=dec2base(a,n);
bb=dec2base(b,n);
nc=char(string(n));
disp([ab ' ' bb ' ' nc])
ct=ct+1;
end
end
end
end
disp(ct)
decimal representation base-n rep.
-------------------------------------------------- ---------
alpha n numerator denominator result alpha
beta beta
1 3 6 3 1.5 2 1 3
2 4 6 8 2.666666667 3 2 4
2 5 10 10 2.5 4 2 5
3 6 10 18 3.6 5 3 6
1 4 12 4 1.333333333 3 1 4
1 6 12 6 1.5 4 1 6
2 8 12 16 2.666666667 6 2 8
6 9 12 54 6.75 8 6 9
3 7 14 21 3.5 6 3 7
4 8 14 32 4.571428571 7 4 8
1 10 15 10 1.666666667 6 1 A
2 6 15 12 2.4 5 2 6
4 12 15 48 4.8 10 4 C
6 10 15 60 6.666666667 9 6 A
1 9 18 9 1.5 6 1 9
2 12 18 24 2.666666667 9 2 C
4 9 18 36 4.5 8 4 9
5 10 18 50 5.555555556 9 5 A
1 5 20 5 1.25 4 1 5
2 10 20 20 2.5 8 2 A
3 12 20 36 3.6 10 3 C
3 15 20 45 3.75 12 3 F
12 16 20 192 12.8 15 C G
2 7 21 14 2.333333333 6 2 7
3 18 21 54 3.857142857 14 3 I
4 14 21 56 4.666666667 12 4 E
10 15 21 150 10.714285714 14 A F
5 11 22 55 5.5 10 5 B
6 12 22 72 6.545454545 11 6 C
1 8 24 8 1.333333333 6 1 8
1 12 24 12 1.5 8 1 C
2 16 24 32 2.666666667 12 2 G
3 9 24 27 3.375 8 3 9
6 18 24 108 6.75 16 6 I
10 16 24 160 10.666666667 15 A G
6 13 26 78 6.5 12 6 D
7 14 26 98 7.538461538 13 7 E
1 21 28 21 1.75 12 1 L
2 8 28 16 2.285714286 7 2 8
3 14 28 42 3.5 12 3 E
4 16 28 64 4.571428571 14 4 G
6 24 28 144 6.857142857 21 6 O
15 21 28 315 15.75 20 F L
1 6 30 6 1.2 5 1 6
1 15 30 15 1.5 10 1 F
1 20 30 20 1.666666667 12 1 K
2 12 30 24 2.4 10 2 C
2 15 30 30 2.5 12 2 F
2 20 30 40 2.666666667 15 2 K
3 10 30 30 3.333333333 9 3 A
3 18 30 54 3.6 15 3 I
4 24 30 96 4.8 20 4 O
6 20 30 120 6.666666667 18 6 K
7 15 30 105 7.5 14 7 F
8 16 30 128 8.533333333 15 8 G
14 21 30 294 14.7 20 E L
20 25 30 500 20.833333333 24 K P
3 22 33 66 3.666666667 18 3 M
4 12 33 48 4.363636364 11 4 C
8 24 33 192 8.727272727 22 8 O
14 22 33 308 14.666666667 21 E M
8 17 34 136 8.5 16 8 H
9 18 34 162 9.529411765 17 9 I
1 14 35 14 1.4 10 1 E
2 28 35 56 2.8 20 2 S
2 30 35 60 2.857142857 21 2 U
6 15 35 90 6.428571429 14 6 F
12 21 35 252 12.6 20 C L
12 30 35 360 12.857142857 28 C U
1 12 36 12 1.333333333 9 1 C
1 18 36 18 1.5 12 1 I
2 9 36 18 2.25 8 2 9
2 24 36 48 2.666666667 18 2 O
4 18 36 72 4.5 16 4 I
5 20 36 100 5.555555556 18 5 K
6 27 36 162 6.75 24 6 R
21 28 36 588 21.777777778 27 L S
77
|
|
Posted by Charlie
on 2022-07-28 16:34:23 |
re(2): computer solution - 3 other solutions
