For positive integers m and n, is m2 - n2 divisible by 3?(1) m - n is divisible by 3(2) When m2 + n2 is divided by 3, the remainder is 2

题目

For positive integers m and n, is m² - n² divisible by 3?

(1) m - n is divisible by 3

(2) When m² + n² is divided by 3, the remainder is 2

选项

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

解析

陈述1： m−n 能被 3 整除 \(m^2 - n^2 = (m + n)(m - n)\) 因此，\(m^2 - n^2\) 一定能被 3 整除条件充分 --- 陈述2：当 \(m^2 + n^2\) 除以 3 时，余数为 2 任何正整数的平方除以 3 时，余数只能是 0 或 1。因此，\(m^2 = 3a + 1\)，\(n^2 = 3b + 1\) \(m^2 - n^2 = 3a + 1 - 3b - 1 = 3(a - b)\) 条件充分 --- {证明：任何正整数的平方除以 3 时，余数只能是 0 或 1} 我们可以将任意正整数表示为 3k、3k+1 或 3k+2 的形式情况1： N = 3k；\(N^2 = 9k^2\) \(N^2\) 除以 3 时余数为 0 情况2： N = 3k+1； \(N^2 = (3k + 1)^2\) = \(9k^2 + 6k + 1\) = \(3(3k^2 + 2k) + 1\) 情况3： N = 3k+2； \(N^2 = (3k + 2)^2\) = \(9k^2 + 12k + 4\) = \(9k^2 + 12k + 3 + 1\) = \(3(3k^2 + 4k + 1) + 1\)

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