h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

When going through h2g2 I sometimes come across hearing 6x9=42 and I understand it is that in base 13 and so it got me thinking about if all the other sums in base whatever to make 42 but it has been a long time since I looked at base in maths. So is there any method in working out what number would equal 42 in base whatever. It probably is such a silly question if there is a simple answer but as I said I haven't looked at base in maths for a long time.

All your base are belong to us?

I have no idea... I'm rubbish with numbers! for being .

I guess that is an ideal question for the Maths Lab A895205

Well 13 is 0xd in hexadecimal and 10 is 0xa, so we could argue that 10 and 13 are the end of the line, but only for Microsoft.

Apologies to anyone who understands this, as well as those of you who don't

Well the number 42 in any other base, say X, converted to base 10 is just 4*X+2. We can create a table in base 10 of the values of this expression for a range of number bases. If you are interested in results of the form A*B=42, then the most useful thing we can have as the next column in our table is the factorisation of the value 42, we'll stick to base 10 for this to make it easy. Then there is a choice of what numbers to have for A and B, since if the value 4*X+2 has more than 2 prime factors, then we can make multiple sums for that case, e.g. to get 12, we can multiply 4*3 or 2*6 - because the factors are 2 (twice) and 3. Finally we can express the numbers back in the same base, X, as we originally used to calculate the value of 42. We start at base 5, which is the minimum base to express the digit 4.

X base10 factors numbers(10) numbers(X)
5 22 2*11 2, 11 2, 21
6 26 2*13 2, 13 2, 21
already we have a pattern, 21 is always a factor in the base X
7 30 2*3*5 (2,15)(3,10)(5,6) (2,21)(3,13)(5,6)
8 34 2*17 2,17 you guessed it 2,21
9 38 2*19 etc
10 42 2*3*7 (2,21)(3,14)(6,7) same as X=10
11 46 2*23 etc
12 50 2*5*5 (2,25)(5,10) (2,21)(5,C)
note, in number bases over 10 we run out of digits, so use alphabet
13 54 2*3*3*3 (2,27)(3,18)(6,9) (2,21)(3,15)(6,9)
14 58 2*29 etc
15 62 2*31 etc
16 66 2*3*11 (2,33)(3,22)(6,11) (2,21)(3,16)(6,B)
17 70 2*5*7 (2,35)(5,14)(7,10) (2,21)(5,E)(7,A)
18 74 2*37 etc
19 78 2*3*13 (2,39)(3,26)(6,13) (2,21)(3,17)(6,D)
20 82 2*41 etc
21 86 2*43 etc
22 90 2*3*3*5 (2,45)(3,30)(5,18)(6,15)(9,10) .....
exercise for reader.

Anyway I haven't checked any of this, so if I've erred let me know.

Oh well, it was nicely formatted before I hit post.