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  3. How many positive integers n have the property that both 3n and n/3 are 4-digit

How many positive integers n have the property that both 3n and n/3 are 4-digit

游客2024-01-13  51
问题 How many positive integers n have the property that both 3n and n/3 are 4-digit integers?
选项 A、111
B、112
C、333
D、334
E、1，134
答案 B
解析 If n is an integer, then 3n is always an integer. Also, 3n will be a 4-digit integer only when 1,000 ≤ 3n ≤ 9,999.Therefore, n is an integer such that ≤ n ≤ 3,333. Equivalently, n is an integer such that 334 ≤ n ≤ 3,333.
If n is an integer, then n/3 is an integer only when n is a multiple of 3. Also, n/3 will be a 4-digit integer only when 1,000 ≤ n/3 ≤ 9,999, or 3,000 ≤ n ≤ 29,997. Therefore, n is a multiple of 3 such that 3,000 ≤ n ≤ 29,997.
It follows that the values of n consist of all multiples of 3 between 3,000 = 3(1,000) and 3,333 = 3(1,111), inclusive. The number of such multiples of 3 is (1,111 - 1,000) + 1 = 112.
The correct answer is B.
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