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 (MDC) e M\303\255nimo M\303\272ltiplo Comum (MMC)

No Maple encontramos alguns comandos para calcularmos o MDC e MMC.

Sejam a e b dois inteiros n\303\243o conjuntamente nulos. Chama-se m\303\241ximo divisor comum de a e b o inteiro positivo d (d>0) que satisfaz \303\240s condi\303\247\303\265es:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIilGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdRJmZhbHNlRicvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSJ+RidGOS9GPEY/Rj0vRkFGP0ZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlUvJStleGVjdXRhYmxlR0Y/Rjk=d \303\251 um divisor comum de a e b, isto \303\251 d |a e d| b; ii) Se o n\303\272mero natural c \303\251 um divisor comum de a e b, ent\303\243o c|d . Em outraspalavras, d \303\251 o maior de todos No Maple utilizamos o comando gcd para calcular o MDC de dois n\303\272meros Exemplo: 1. Calcular o MDC (24, 6) R:

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IiIn

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFH

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEkZ2NkRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictSSNtbkdGJDYkUSUxNTUwRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjs2JFEkMjAwRidGPi8lK2V4ZWN1dGFibGVHRkZGPkY+RmZuRj4=

IiNd

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRKm51bXRoZW9yeUYnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZALUkjbW9HRiQ2LVEiOkYnRkAvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGPy8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZWRj1GQA==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSREMjRHRig8KiIiIiIiIyIiJCIiJSIiJyIiKSIjNyIjQzcjRi4=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JVEjRDZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSlkaXZpc29yc0YnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYlLUkjbW5HRiQ2JFEiNkYnRjkvJStleGVjdXRhYmxlR0Y9RjlGOS1GNjYtUSI7RidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnRk0tRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZbby9GTkZcby1GNjYtUSIjRidGOUY7Rj5GQEZCRkRGRkZIRltvRmBvLUYsNiVRL0Rpdmlzb3Jlc35kZX42RidGL0YyRmVuRjk=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNENkdGKDwmIiIiIiIjIiIkIiInNyNGLg==

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LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS1JbnRlcnNlfF53fGJ1fF53fF51b0dGKDwmIiIiIiIjIiIkIiInNyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRNRENHRigiIic3I0Yu

Para calcular o MDC de dois ou mais n\303\272meros fazemos uso do comando igcd ou o fato de que mdc (a,b,c) = (mdc(a,b),c) JSFH

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IiIm

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

IiIm

JSFH

M\303\255nimo M\303\272ltiplo Comum (MMC) Sejam a e b dois inteiros diferentes de zero. Chama-se m\303\255nimo m\303\272ltiplo comum (mmc) de a e b o inteiro m(m>0) que satisfaz \303\240s condi\303\247\303\265es: i) a|m e b| m ii) se a|c e se b| c, c>0, ent\303\243o m \303\251 menor ou igual a c No Maple utilizamos o comando lcm para calcular o mmc entre dois ou mais n\303\272meros. Exemplo: 1.Calcular o MMC (3,25) R:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1JI21vR0YkNi1RIn5GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkYvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUnbHNwYWNlR0Y3LyUncnNwYWNlR0Y3LyUrZXhlY3V0YWJsZUdGRkZBNo exemplo acima, constru\303\255mos o conjuntos dos 50 primeiros m\303\272ltiplos positivos de 3 e chamamos de M3; depois constru\303\255mos o conjunto dos 50 primeiros m\303\272ltiplos de 25 e chamamos de M25. Em seguida, fizemos a interse\303\247\303\243o desses dois conjuntos encontrando m\303\272ltiplos comuns de 3 e 25; Vemos que 3 e 25 dividem todos os elementos de M6interM25, o que corresponde a primeira propriedade, e pela propriedade ii) chegamos a conclus\303\243o que o mmc(3,25)= 75, pois \303\251 o menor dos elementos da interse\303\247\303\243o dos conjuntos dos m\303\272ltiplos.