OEF ÕûÊý·Ö½â --- ½éÉÜ ---

±¾Ä£¿éÄ¿Ç°°üº¬ 14 ¸ö¹ØÓÚÕûÊýÒòʽ·Ö½âµÄ³õµÈÁ·Ï°: ´æÔÚÐÔ, ΨһÐÔ, ¹ØÓÚ gcd ºÍ lcm µÄ¹ØϵµÈ.

Òò×ÓÊý

¸ø³öÒ»¸öÕûÊý ËüÇ¡ÓÐ ¸öÒò×Ó (1 ºÍ Ò²ÊÇ µÄÒò×Ó) ¶øÇÒËüÖÁÉÙ±» ¸ö²»Í¬ËØÊý³ý¾¡.

ÕûÊýµÄÒò×Ó

ÉèÕûÊý Ç¡ÓÐ 3 ¸ö²»Í¬ËØÊýÒò×Ó :

ÒÑÖª ±È ¶à ¸öÒò×Ó, ²¢ÇÒ ±È ¶à ¸öÒò×Ó.

¸ø³ö µÄËùÓпÉÄÜÐÔ : Çë°´ µÄÔö³¤´ÎÐòÒ»ÐÐдһ¸ö½â , , (ÒÔ¶ººÅ·Ö¸ô).


Òò×Ó

ÓÐÒ»¸öÕûÊý , ÆäÒòʽ·Ö½âΪ

= .

ÒÑÖª Õû³ý , µÈÓÚʲô?


Òòʽ·Ö½âÖ®ºÍ

Éè ºÍ ÊÇÁ½¸öÕý, Ëü¾ßÓÐÈçÏÂÒòʽ·Ö½â:

= 123 , = 124 ,

ÆäÖÐÒò×Ó i ÊDz»Í¬µÄËØÊý.

ÊÇ·ñ¿ÉÄÜÓÐÐÎÈç

| | = 123 ,

µÄ·Ö½âÆäÖÐ i ÊDz»Í¬µÄËØÊý?


ÇóÒò×Ó II

ÓÐÁ½¸öÕûÊýµÄËØÒò×Ó·Ö½âʽ:

=    ,    = ,

ÆäÖÐÒò×Ó , ÊDz»Í¬ËØÊý. ÇóÕâЩÒò×Ó.


ÇóÒò×Ó III

ÓÐÁ½¸öÕûÊýµÄËØÒò×Ó·Ö½âʽ:

=    ,    = ,

ÆäÖÐÒò×Ó , , ÊDz»Í¬ËØÊý. ÇóÕâЩÒò×Ó.


×î´ó¹«ÒòÊý

Éè m, n ÊÇÓÐÒÔÏÂÒòʽ·Ö½âʽµÄÁ½¸öÕýÕûÊý.

m = , n = ,

ÆäÖÐ , , ÊDz»Í¬ËØÊý.

¼ÆËã gcd(m,n) ×÷Ϊ , , µÄº¯Êý.


×îС¹«±¶Êý

Éè m, n ÊÇÓÐÒÔÏÂÒòʽ·Ö½âʽµÄÁ½¸öÕýÕûÊý.

m = , n = ,

ÆäÖÐ , , ÊDz»Í¬ËØÊý.

¼ÆËã lcm(m,n) ×÷Ϊ , , µÄº¯Êý.


ËØÒò×Ó×î¶à¸öÊý

Éè ÊÇ Î»ÕûÊý. ÒÑÖª ûÓÐ < µÄËØÒò×Ó, ×î¶à¿ÉÄÜÓжàÉÙ¸öËØÒò×Ó ?

Òò×ÓÊý II

Éè ÊǾßÓÐÒÔϵ½²»Í¬ËØÒò×Ó·Ö½âµÄÕýÕûÊý.

= 1 2

µÄÒò×Ó¸öÊýÊǶàÉÙ ? ( µÄÒò×ÓÊdzý¾¡ µÄÕýÕûÊý, °üÀ¨ 1 Óë ÔÚÄÚ.)


Òò×ÓÊý III

Éè ÊǾßÓÐÒÔϵ½²»Í¬ËØÒò×Ó·Ö½âµÄÕýÕûÊý.

= 1 2 3

µÄÒò×Ó¸öÊýÊǶàÉÙ ? ( µÄÒò×ÓÊdzý¾¡ µÄÕýÕûÊý, °üÀ¨ 1 Óë ÔÚÄÚ.)


ÊÔ³ý

ÎÒÃÇÓÐÒ»¸öÕûÊý < , ¶øÇÒÒªÓÃÖð´ÎÒÔ 2,3,4,5,6,... ÊÔ³ý µÄ·½·¨ÕÒ³ö µÄËØÒò×Ó. ÒÑÖª ÓÐÒ»¸öËØÒò×Ó·Ö½â

= 11 22 ... tt

ÆäÖÐÖ¸ÊýÖ®ºÍ 1+2+...+t = , (µ«ÊÇÒò×Ó i ²»ÖªµÀ) ÄÇôÔÚ×µÄÇéÐÎÄãÒªÊÔ³ýµÄ×î´óÒò×ÓÊǶàÉÙ (²»Ð迼ÂÇ´ËÒò×ÓÊÇ·ñËØÊý) ?


Á½¸öÒò×Ó

¼ÆËã¾ßÓÐÐÎΪ

= × ,

µÄËØÒò×Ó·Ö½âµÄÕýÕûÊý < µÄ¸öÊý, ÕâÀïµÄÖ¸Êý ºÍ ÊÇ > µÄÕûÊý.


Á½¸öÒò×Ó II

¼ÆËã¾ßÓÐÐÎΪ

= × ,

µÄËØÒò×Ó·Ö½âµÄÕýÕûÊý < µÄ¸öÊý, ÕâÀïµÄÖ¸Êý ºÍ ÊÇ > µÄÕûÊý.

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