GCD (最大公因数)的性质

10.GCD (最大公因数)的性质2024-08-08

GCD 的性质总结

$g c d (a_{1}, a_{2}, . . . . . . a_{n}) = g c d (| a_{1} |, | a_{2} |, . . . . . . | a_{n} |)$
$g c d (a, 0) = g c d (a, a) = | a |$
$g c d (a_{1}, a_{2}, . . . . . . a_{n - 1}, a_{n}) = g c d (g c d (a_{1}, a_{2}), a_{3}, . . . . . . a_{n - 1}, a_{n})$
对于不全为零的整数 $a_{1}, a_{2}, a_{3}, . . . . . . a_{n - 1}, a_{n}, \forall 1 < k < n - 1$ , $g c d (a_{1}, a_{2}, a_{3}, . . . . . . a_{n - 1}, a_{n}) = g c d (g c d (a_{1}, a_{2}, a_{3}, . . . a_{k}), g c d (a_{k + 1}, . . ., a_{n}))$
对于不全为零的整数 $a_{1}, a_{2}, a_{3}, . . . . . . a_{n - 1}, a_{n}$ , $g c d (m \cdot a_{1}, m \cdot a_{2}, . . . m \cdot a_{n}) = | m | \cdot g c d (a_{1}, a_{2}, a_{3}, . . . . . . a_{n - 1}, a_{n})$
对于不全为零的整数 $a_{1}, a_{2}, a_{3}, . . . . . . a_{n - 1}, a_{n}$ ，若 $g c d (a_{1}, a_{2}, a_{3}, . . ., a_{n}) = d$ , 则有 $g c d (\frac{a_{1}}{d}, \frac{a_{2}}{d}, \frac{a_{3}}{d}, . . ., \frac{a_{n}}{d}) = 1$
$g c d (a_{1}, a_{2}, . . ., a_{n}, d) \leq d$ ,特别的 $g c d (a_{1}, a_{2}, . . ., a_{n}, 1) = 1$ , 且 $g c d (a_{1}, a_{2}, . . ., a_{n}) = 1$ 且 $a_{i}! = 1$ 是 $a_{1}, a_{2}, . . ., a_{n}$ 中至少存在一对 $a_{i}, a_{j}$ 互质的充分必要条件.