Problem 15 · 2010 Math Kangaroo
Medium
Number Theory
digit-sumcareful-counting
How many whole numbers are there whose digits sum to 2010 and have a product of 2?
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Answer: B — 2009
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Hint 1 of 2
If the digits multiply to 2, what can the digits be?
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Hint 2 of 2
Fix the digits, then count where the single 2 can go.
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Approach: digit-product forces the digits
- A digit product of 2 means one digit is 2 and all the rest are 1's.
- For the digits to add to 2010 we need 2 plus 2008 ones, so 2009 digits in all.
- The single 2 can sit in any of the 2009 places, giving 2009 numbers.
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