Problem 18 · 2018 Math Kangaroo
Hard
Number Theory
place-value
Instead of digits, Hannes uses the letters A, B, C, and D in a calculation. Different letters stand for different digits, and the addition ABC + CBA = DDDD is correct (each group of letters is a number written with those digits in order). Which digit does the letter B stand for?
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Answer: A — 0
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Hint 1 of 2
Add ABC and CBA column by column and notice the symmetry of the outer digits.
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Hint 2 of 2
The sum 101(A + C) + 20B must equal a four-digit repdigit DDDD; only D = 1 fits, which pins B.
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Approach: add by place value and match to a repdigit
- ABC + CBA = 100(A+C) + 20B + (A+C) = 101(A+C) + 20B.
- This must equal DDDD = 1111 x D, and since the sum is below 2000 we need D = 1, so 101(A+C) + 20B = 1111.
- Taking A + C = 11 gives 1111, leaving 20B = 0.
- So B stands for 0.
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