Problem 17 · 2024 Math Kangaroo
Hard
Number Theory
factorization
A teacher writes the 7 digits 1 1 2 2 2 2 3 on the board. He asks a student to insert some multiplication signs (×) in such a way that the product of the resulting numbers (possibly with multiple digits) has the value 2024. How many multiplication signs must be inserted?
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Answer: D — 4
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Hint 1 of 2
Factor 2024 first to see which building-block numbers the digits must form.
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Hint 2 of 2
2024 = 11 × 23 × 8, so the digits must group into 11, 23 and the factor 8 from three 2s.
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Approach: factor the target and group the digits
- 2024 = 2³ × 11 × 23.
- Keeping the digits 1 1 2 2 2 2 3 in order, form 11, then three single 2s, then 23.
- So 11 × 2 × 2 × 2 × 23 = 2024 uses five numbers, needing 4 multiplication signs.
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