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I don't see why A=B=C=0 is not a possible solution.
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I know it's not stated in the question, but if A=B=C=0 then what's the point of writing A B C + A B C + A B C = C C C. You can just rewrite all with C C C or A A A or B B B.
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The real question is "why not?". Perhaps this question was meant to encompass this additional bit of complexity. There's essentially no problem in having A=B=C=0, even if you can write it in a simpler way.
@Julie Foster In a cryptogram,by definition,all the letters stand for distinct digits
I think this is the best solution. :)
This explanation is more conclusive from result obtained than verification with careful steps. Nevertheless, a half guess is also a good way to answer questions in Brilliant.
3 ( 1 0 0 a + 1 0 b + c ) = 1 1 1 c , or simply 1 0 a + b = 5 1 8 c . Here we will notice that between 1 to 9, the only possible solution is that ( a , b , c ) = ( 1 , 8 , 5 ) , thus, a b c = 1 8 5 .
3C =XC : Only solution is, C=5 ( 3 × 5 = 1 5 )
Reminder, X = 1
3B+1(reminder) = X5
i.e 3B = X4 Only solution is, B 8 ( 3 × 8 = 2 4 )
Reminder, X = 2
3A+2(reminder) = 5
i.e 3C = 3. Only solution is, A=1
Hence A B C = 1 8 5 .
As A = B = C = 0 is not where we should cease particularly when we know that it is usual to have different digits being meant by different alphabets' representation, we should look for others.
Among 1 to 9, only 5 added itself consecutively for twice such as multiplied by 3 which shall produce digit at 1 0 0 of its own, therefore, C = 5.
With C = 5, it carries forward a 1 to add onto another digit multiplied by 3. Only 8 × 3 + 1 shall give 1 0 1 of 5 among 0 to 9, therefore, B = 8.
With C = 5 and B = 8, it carries forward a 2 onto another digit (different) multiplied by 3. Only 1 × 3 + 2 give 1 0 2 of 5 among 1 to 9 (first digit is usually meant for non-zero), therefore, A = 1.
850 × 3 = 2550 is obviously not the answer, so, 1 0 0 or C not considered for 0 is correct.
Therefore, 185 is the only answer.
Answer: 1 8 5
Assume A, B, and C are single digit integers. The highest number one of these can be is 9. From the first column: 3C = n(10)+C. Because 3(9)=27, n can't be greater than 2. Therefore, there are three options:
3
C
=
C
3
C
=
1
0
+
C
3
C
=
2
0
+
C
Solving each, one gets:
C
=
0
C
=
5
C
=
1
0
5 is the only single digit; therefore, C=5.
The second column now is evaluating 3B+1 because you have to "carry the one" you get by getting 15 after adding 5+5+5. By that logic:
3
B
+
1
=
5
3
B
+
1
=
1
5
3
B
+
1
=
2
5
Evaluating each, one gets:
B
=
4
/
3
B
=
1
4
/
3
B
=
8
B is the only integer; therefore, B=8.
The sum of the last column has to be exactly 5 because there aren't any more columns after that where you can "carry the one" to. By the same logic as for the second column, the last column is evaluating 2+3A. Therefore:
2
+
3
A
=
5
A
=
1
If C=5, B=8 and A=1, then ABC=185.
185 + 185 + 185 = 555; then ABC = 185
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Only solution where C+C+C=C is possible is if C=5.
Since the final answer is CCC i.e., 555, and they are 3 equal numbers which add to it, then just divide 555 by 3:
555÷3=185