Problem 5 · 2011 Math Kangaroo
Medium
Number Theory
divisibility
If 2011 is divided by a certain positive whole number, the remainder is 1011. Which number was it divided by?
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Answer: E — There is no such number.
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Hint 1 of 2
The remainder must be smaller than the divisor, so the divisor exceeds 1011.
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Hint 2 of 2
Write 2011 = (divisor)×(quotient) + 1011 and see what the divisor would have to divide.
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Approach: use that remainder < divisor
- If the remainder is 1011, the divisor must be larger than 1011.
- Then 2011 = d·q + 1011, so d·q = 1000, meaning d divides 1000 and d ≤ 1000.
- A divisor cannot be both greater than 1011 and at most 1000.
- So there is no such number, choice (E).
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