A friend of mine sent me this puzzle.
A,B,C are all different digits. The following is a sum.
A,B,C are all different digits. The following is a sum.
ABC
ABC
ABC
--------
CCC
Identify the three digits.
Also solve for A, B, C if it's a hexadecimal sum (base 16).
Base 16
1=1
2=2
3=3
4=4
5=5
6=6
7=7
8=8
9=9
10=A
11=B
12=C
13=D
14=E
15=F
Base 16
1=1
2=2
3=3
4=4
5=5
6=6
7=7
8=8
9=9
10=A
11=B
12=C
13=D
14=E
15=F
Solution for base 16
https://docs.google.com/spreadsheets/d/12w8NwIwliFBaJ9h2YnityTWUKrL2zRtcGVaE8Qm4rfg/edit?usp=drivesdk
Another method:
Each of the numbers is
256x+16y+z
3*(256x+16y+z)
=768x+48y+3z
The sum with all 3 digits as z
=256z+16z+z=273z
Thus, 768x+48y=270z
128x+8y=45z
Z has to be an even number. More important since the LHS is a multiple of 8, z also has to be a multiple of 8. The max vape of x, y, z is 16-1=15. Hence z can only be 8.
128+8y=45*8
16x+y=45
X=2, y=13, z=8
Another method:
Each of the numbers is
256x+16y+z
3*(256x+16y+z)
=768x+48y+3z
The sum with all 3 digits as z
=256z+16z+z=273z
Thus, 768x+48y=270z
128x+8y=45z
Z has to be an even number. More important since the LHS is a multiple of 8, z also has to be a multiple of 8. The max vape of x, y, z is 16-1=15. Hence z can only be 8.
128+8y=45*8
16x+y=45
X=2, y=13, z=8
It is interesting that there seems to be only one solution each for bases 10, 16. And no solution for other bases from 5 to 16.
Additional Reading
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