If a and b are consecutive positive integers such that a2 is divisible by 25 and b2 is divisible by 63, which of the following is the greatest integer that must be a factor of ab ?

题目

If a and b are consecutive positive integers such that a² is divisible by 25 and b² is divisible by 63, which of the following is the greatest integer that must be a factor of ab ?

选项

105

210

225

315

解析

如果\(a\)和\(b\)是连续的正整数，使得\(a^{2}\)能被\(25\)整除，\(b^{2}\)能被\(63\)整除，那么以下哪个是必定为\(ab\)因数的最大整数？ \(a^2\) 能被 25（\(5\times5\)）整除，因此 \(a\) 必含因数 5。 \(b^2\) 能被 63（\(3\times3\times7\)）整除，因此 \(b\) 必含因数 21。若 \(a\)、\(b\) 为连续整数，则其中一个必为偶数，另一个必为奇数。 \[ 5\times21\times2=210 \] 答案：C

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