Problem 11 · 2019 Math Kangaroo
Medium
Number Theory
place-valuesum-constraint
Three four-digit numbers are written on three separate strips of paper, as shown. The sum of the three numbers is 10126. Three of the digits in the picture are hidden. Which are the hidden digits?
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Answer: A — 5, 6 and 7
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Hint 1 of 3
You don't need to find each number — just the total of the three hidden digits.
Still stuck? Show hint 2 →
Hint 2 of 3
The digit sum of a number leaves the same remainder on division by 9 as the number itself, so the digits of all three numbers together must match 10126 in that test.
Still stuck? Show hint 3 →
Hint 3 of 3
Add the nine visible digits, see what the three hidden ones must add to, then pick the option with that sum.
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Approach: use the divisible-by-9 (digit-sum) check instead of reconstructing the numbers
- A number and its digit sum leave the same remainder when divided by 9, so the sum of all twelve digits leaves the same remainder as 10126 does.
- 10126 has digit sum 1+0+1+2+6 = 10, which leaves remainder 1; the nine visible digits add to 1+2+4+3+7+2+1+2+6 = 28, which also leaves remainder 1.
- So the three hidden digits must add to a multiple of 9; among the choices only 5+6+7 = 18 works.
- The hidden digits are 5, 6 and 7.
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