10*12^5 + 7*12^4 + x*12^3 + 11*12^2 + 5*12 + y is to be divisible by 8*12+3, which is 99.
Without the terms involving x and y:
10*12^5 + 7*12^4 + 11*12^2 + 5*12 = 2635116, which is congruent to 33 mod 99.
x*12^3 + y must be congruent to 66 mod 99. Of course, in addition each of x and y must be in the range 0 to 11.
12^3 = 1728 is congruent to 45 mod 99. If we try to leave well enough alone and let x=1 and make y congruent to 21 mod 99 we won't succeed in getting a value in the proper range.
Clearly x=0 won't work either as that would need y congruent to 66 mod 99.
We need to try every x to see what y is needed for that x.
Well what use is a computer if you don't use it for the grunt work. Without writing a script (program), just type:
>> for x= 1:11 y=mod(66-45*x,99); disp([x y])
end
1 21
2 75
3 30
4 84
5 39
6 93
7 48
8 3
9 57
10 12
11 66
Clearly x=8 and y=3.
Verified with
>> mod(base2dec('a78b53',12),base2dec('83',12))
ans =
0
BTW, when divided by 83 base 12 the answer is 13599 base 12.
In base 10 that's 2648943/99 = 26757
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Posted by Charlie
on 2022-08-04 09:31:39 |
solution
