Problem 24 · 2021 Math Kangaroo
Stretch
Number Theory
Algebra & Patterns
place-valuedigit-sum
The 6-digit number 2ABCDE is multiplied by 3 and the result is the 6-digit number ABCDE2. What is the sum of the digits of this number?
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Answer: B — 27
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Hint 1 of 2
Let X be the five-digit block ABCDE; write both 2ABCDE and ABCDE2 using X.
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Hint 2 of 2
Set up the equation from 'times 3' and solve for X, then add its digits with the 2.
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Approach: turn the digit-shuffle into one equation
- 2ABCDE = 200000 + X and ABCDE2 = 10X + 2, where X = ABCDE.
- 3(200000 + X) = 10X + 2 gives 7X = 599998, so X = 85714 and the number is 285714.
- Its digits sum to 2+8+5+7+1+4 = 27.
- So the answer is B.
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