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.
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=a*b;
denominator=a+b/n;
f=numerator/denominator;
if round(f)==f
fprintf('%2d %2d %2d %13.9f %13.9f %13.9f\n',a, b, n, numerator, denominator, f)
ct=ct+1;
end
end
end
end
disp(ct)
for n=2:36
for a=1:n-1
for b=1:n-1
numerator=a*b;
denominator=a+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)
The output of the bottom half was placed next to that of the top, deleting the duplicate column, to get:
decimal representation base-n rep.
-------------------------------------------------- ---------
alpha n numerator denominator result alpha
beta beta
1 3 6 3.000000000 1.500000000 2.000000000 1 3
2 4 6 8.000000000 2.666666667 3.000000000 2 4
2 5 10 10.000000000 2.500000000 4.000000000 2 5
3 6 10 18.000000000 3.600000000 5.000000000 3 6
1 4 12 4.000000000 1.333333333 3.000000000 1 4
1 6 12 6.000000000 1.500000000 4.000000000 1 6
2 8 12 16.000000000 2.666666667 6.000000000 2 8
6 9 12 54.000000000 6.750000000 8.000000000 6 9
3 7 14 21.000000000 3.500000000 6.000000000 3 7
4 8 14 32.000000000 4.571428571 7.000000000 4 8
2 6 15 12.000000000 2.400000000 5.000000000 2 6
4 12 15 48.000000000 4.800000000 10.000000000 4 C
6 10 15 60.000000000 6.666666667 9.000000000 6 A
1 9 18 9.000000000 1.500000000 6.000000000 1 9
2 12 18 24.000000000 2.666666667 9.000000000 2 C
4 9 18 36.000000000 4.500000000 8.000000000 4 9
5 10 18 50.000000000 5.555555556 9.000000000 5 A
1 5 20 5.000000000 1.250000000 4.000000000 1 5
2 10 20 20.000000000 2.500000000 8.000000000 2 A
3 12 20 36.000000000 3.600000000 10.000000000 3 C
3 15 20 45.000000000 3.750000000 12.000000000 3 F
12 16 20 192.000000000 12.800000000 15.000000000 C G
2 7 21 14.000000000 2.333333333 6.000000000 2 7
3 18 21 54.000000000 3.857142857 14.000000000 3 I
4 14 21 56.000000000 4.666666667 12.000000000 4 E
5 11 22 55.000000000 5.500000000 10.000000000 5 B
6 12 22 72.000000000 6.545454545 11.000000000 6 C
1 8 24 8.000000000 1.333333333 6.000000000 1 8
1 12 24 12.000000000 1.500000000 8.000000000 1 C
2 16 24 32.000000000 2.666666667 12.000000000 2 G
3 9 24 27.000000000 3.375000000 8.000000000 3 9
6 18 24 108.000000000 6.750000000 16.000000000 6 I
10 16 24 160.000000000 10.666666667 15.000000000 A G
6 13 26 78.000000000 6.500000000 12.000000000 6 D
7 14 26 98.000000000 7.538461538 13.000000000 7 E
1 21 28 21.000000000 1.750000000 12.000000000 1 L
2 8 28 16.000000000 2.285714286 7.000000000 2 8
3 14 28 42.000000000 3.500000000 12.000000000 3 E
4 16 28 64.000000000 4.571428571 14.000000000 4 G
6 24 28 144.000000000 6.857142857 21.000000000 6 O
15 21 28 315.000000000 15.750000000 20.000000000 F L
1 6 30 6.000000000 1.200000000 5.000000000 1 6
1 15 30 15.000000000 1.500000000 10.000000000 1 F
2 12 30 24.000000000 2.400000000 10.000000000 2 C
2 15 30 30.000000000 2.500000000 12.000000000 2 F
2 20 30 40.000000000 2.666666667 15.000000000 2 K
3 10 30 30.000000000 3.333333333 9.000000000 3 A
3 18 30 54.000000000 3.600000000 15.000000000 3 I
4 24 30 96.000000000 4.800000000 20.000000000 4 O
6 20 30 120.000000000 6.666666667 18.000000000 6 K
7 15 30 105.000000000 7.500000000 14.000000000 7 F
8 16 30 128.000000000 8.533333333 15.000000000 8 G
14 21 30 294.000000000 14.700000000 20.000000000 E L
20 25 30 500.000000000 20.833333333 24.000000000 K P
3 22 33 66.000000000 3.666666667 18.000000000 3 M
4 12 33 48.000000000 4.363636364 11.000000000 4 C
8 24 33 192.000000000 8.727272727 22.000000000 8 O
14 22 33 308.000000000 14.666666667 21.000000000 E M
8 17 34 136.000000000 8.500000000 16.000000000 8 H
9 18 34 162.000000000 9.529411765 17.000000000 9 I
1 14 35 14.000000000 1.400000000 10.000000000 1 E
2 28 35 56.000000000 2.800000000 20.000000000 2 S
2 30 35 60.000000000 2.857142857 21.000000000 2 U
6 15 35 90.000000000 6.428571429 14.000000000 6 F
12 21 35 252.000000000 12.600000000 20.000000000 C L
12 30 35 360.000000000 12.857142857 28.000000000 C U
1 12 36 12.000000000 1.333333333 9.000000000 1 C
1 18 36 18.000000000 1.500000000 12.000000000 1 I
2 9 36 18.000000000 2.250000000 8.000000000 2 9
2 24 36 48.000000000 2.666666667 18.000000000 2 O
4 18 36 72.000000000 4.500000000 16.000000000 4 I
5 20 36 100.000000000 5.555555556 18.000000000 5 K
6 27 36 162.000000000 6.750000000 24.000000000 6 R
21 28 36 588.000000000 21.777777778 27.000000000 L S
74 results
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Posted by Charlie
on 2022-07-27 11:51:44 |
computer solution
