设[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex],[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]是整数且不全为0，而[tex=9.857x1.214]hhHzRVDsWGXE+Yltfe39hDUdsl3Yzf9jGRPDg4wYEoJYR6eBGAfms1GUG8a2PN1l[/tex]，证明[tex=0.571x1.0]TcM6B5Wrs5vy9dWrxRPSdg==[/tex]是[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]与[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的一个最大公因数当且仅当[tex=4.214x1.357]jI1oqbiyUHYU1xbNvvBdDK5ib01K7Vb7AmVkL7RKEyk=[/tex]

2022-06-08

设[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex],[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]是整数且不全为0，而[tex=9.857x1.214]hhHzRVDsWGXE+Yltfe39hDUdsl3Yzf9jGRPDg4wYEoJYR6eBGAfms1GUG8a2PN1l[/tex]，证明[tex=0.571x1.0]TcM6B5Wrs5vy9dWrxRPSdg==[/tex]是[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]与[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的一个最大公因数当且仅当[tex=4.214x1.357]jI1oqbiyUHYU1xbNvvBdDK5ib01K7Vb7AmVkL7RKEyk=[/tex]

答案：

因[tex=1.286x1.214]rkgrF+YaaESwSQDjR6KfWg==[/tex]不全为0，且[tex=5.5x1.214]w72Ofn7/hsqgiC7yrIeYmB6xWNOLzuhTDZotAB0omek=[/tex]，所以[tex=2.357x1.214]DazOm8k4LtdmCYWhQdVUTw==[/tex]. 充分性 由于[tex=5.5x1.214]6E2ZU16dtsas8E49OBK5us64LZrRiEamV6ZAisE8eZY=[/tex]，[tex=0.571x1.0]TcM6B5Wrs5vy9dWrxRPSdg==[/tex]是[tex=1.286x1.214]rkgrF+YaaESwSQDjR6KfWg==[/tex]的公因数. 若[tex=4.214x1.357]KDZrtWBuBbuLDtgLcpVdREiP8twPXhianiA5cqSTRKk=[/tex]，则[tex=3.071x1.214]Q1LDwSYUyNw6N1V5PfuldCPYMOaQkQEPAZwjIf/BMgk=[/tex]，使[tex=4.786x1.214]wS9L5EeNDTbMlyXiLfONBZ4qeZI8dMeVcGevgoP5z2A=[/tex]，两边同时乘以[tex=0.571x1.0]TcM6B5Wrs5vy9dWrxRPSdg==[/tex]得[tex=4.143x1.143]HomSScy2MD3aedi57ym3UQ==[/tex].于是对[tex=1.286x1.214]rkgrF+YaaESwSQDjR6KfWg==[/tex]的任何公因数[tex=3.5x1.357]4zAj4d0t+6fDShwdrZaldQ==[/tex]，必有[tex=2.071x1.357]0XT6ZSN3SCIAAlUaerBD6Q==[/tex]故[tex=0.571x1.0]TcM6B5Wrs5vy9dWrxRPSdg==[/tex]为[tex=1.286x1.214]rkgrF+YaaESwSQDjR6KfWg==[/tex]最大公因数.必要性 若[tex=0.571x1.0]TcM6B5Wrs5vy9dWrxRPSdg==[/tex]为[tex=1.286x1.214]rkgrF+YaaESwSQDjR6KfWg==[/tex]最大公因数.则[tex=3.071x1.214]n7qzr7ufo4LkB94Jc6ZH8qOcxzPUP5EqV+9HBl+rJ+8=[/tex],使[tex=4.143x1.143]a8wbjcmYTcwjFKxTTurBYQ==[/tex]，两边同时除以[tex=0.571x1.0]TcM6B5Wrs5vy9dWrxRPSdg==[/tex]，得[tex=4.786x1.214]wS9L5EeNDTbMlyXiLfONBZ4qeZI8dMeVcGevgoP5z2A=[/tex]，从而[tex=4.214x1.357]YiPNsyTscVc9qLEGI9EPcC5XfdSAcIjPJeLp4SpgcvM=[/tex]

举一反三

内容

0
图9-26所示，金属球[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]和金属球壳[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]同心放置，它们原先都不带电。设球为[tex=0.857x1.0]6WwbFXETRyeyXlvAruSoNg==[/tex]，球壳[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的内、外半径分别为[tex=0.857x1.0]BNzznGkXRFuGyw2vMy6rWw==[/tex]和[tex=0.857x1.0]Fz01PbYkU0SRGm3tB5KjiA==[/tex]。求在下列情况下[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]、[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的电势差：使[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]带[tex=1.286x1.143]HtShgpkNmCs66SDQIt6cYg==[/tex],使[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]带[tex=1.286x1.143]NeILClIc8twxeO5vjYm8Dw==[/tex][img=203x220]17e1aa8f4234a38.png[/img]
1
图9-26所示，金属球[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]和金属球壳[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]同心放置，它们原先都不带电。设球为[tex=0.857x1.0]6WwbFXETRyeyXlvAruSoNg==[/tex]，球壳[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的内、外半径分别为[tex=0.857x1.0]BNzznGkXRFuGyw2vMy6rWw==[/tex]和[tex=0.857x1.0]Fz01PbYkU0SRGm3tB5KjiA==[/tex]。求在下列情况下[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]、[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的电势差：使[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]带[tex=1.286x1.143]NeILClIc8twxeO5vjYm8Dw==[/tex],将[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的外表面接地。[img=203x220]17e1aa8f4234a38.png[/img]
2
求向量[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]与[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的夹角：[tex=5.429x1.357]28vNNArDuRRnb+06piGt7g==[/tex]，[tex=6.143x1.357]Uqwu1QGYB2waFjV52ImPKg==[/tex] .
3
静止型[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]地中海贫血患者之间婚配，生出轻型[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]地中海贫血患者的可能性是 A: 0 B: 1/8 C: 1/4 D: 1/2 E: 1
4
求向量[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]与[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的夹角：[tex=5.429x1.357]2a1nHnVLv213QxcHtU3H+w==[/tex]，[tex=5.357x1.357]1D0C8TkZZ+4Sbb/nx0LsXA==[/tex] .