Problem 26 · 2020 Math Kangaroo
Stretch
Number Theory
divisibilitydigit-sum
A three-digit number is called balanced if its middle digit is the average of the other two digits. How many balanced numbers are divisible by 18?
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Answer: C — 6
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Hint 1 of 2
Divisible by 18 means even and digit-sum divisible by 9; 'balanced' means the middle is the average of the outer two.
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Hint 2 of 2
Balanced makes the digit sum three times the middle digit, so the middle digit must be a multiple of 3.
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Approach: combine the balanced condition with divisibility by 18
- Balanced means middle = (first+last)/2, so the digit sum first+middle+last = 3 x middle.
- Divisible by 9 needs that sum divisible by 9, so the middle digit is 0, 3, 6 or 9; divisible by 2 needs an even last digit.
- Listing the cases gives 234, 432, 468, 630, 666, 864 - exactly 6 numbers.
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