Problem 22 · 2009 Math Kangaroo
Stretch
Counting & Probability
divisibilitycareful-counting
Let Z be the number of 8-digit numbers that are made up of all different digits, none equal to 0. How many of those numbers are divisible by 9?
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Answer: C — Z9
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Hint 1 of 2
By symmetry, the eight-digit numbers split evenly by their remainder when divided by 9.
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Hint 2 of 2
Figure out how many of the nine possible remainders these numbers can have, then take the equal share.
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Approach: split the count evenly across remainders mod 9
- Each number uses eight distinct nonzero digits; permuting digits spreads the values evenly over the nine remainders mod 9.
- So exactly one ninth of them are divisible by 9.
- That share is Z/9.
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