Problem 8 · 2013 Math Kangaroo
Medium
Number Theory
divisionoff-by-one
The number n is the biggest natural number for which \(4n\) is three digits long, and m is the smallest natural number for which \(4m\) is three digits long. What value does \(4n - 4m\) have?
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Answer: C — 896
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Hint 1 of 2
A three-digit multiple of 4 ranges from 100 to 999; find the largest and smallest n that land in that range.
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Hint 2 of 2
Then just subtract the two products.
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Approach: bound the three-digit multiples of 4
- Largest n with 4n ≤ 999: n = 249, so 4n = 996.
- Smallest m with 4m ≥ 100: m = 25, so 4m = 100.
- Then 4n − 4m = 996 − 100 = 896.
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