T. J. R o’rinli bo’lganda ni toping ?

4 Qoldiqni hisoblashning effektiv usulidan foydalanib ni hisoblang

Qoldiqni hisoblashning effektiv usulidan foydalanib 91^746 (mod 75) ni hisoblashimiz kerak. Ushbu usul bilan, qoldiqni hisoblashni osonlashtirishimiz mumkin.

1. 
   91^746 (mod 75) ni topish uchun 91 ni 75 ga qarab ko'ramiz: 91 mod 75 = 16.
   
2. 
   91^746 ni hisoblash uchun 75 ga qoldiqsiz ko'paytirib boramiz. Buning uchun keyingi sonlar ketma-ketlikni o'z ichiga oladi:
   

91^746 ≡ 91^(2^0) * 91^(2^1) * 91^(2^2) * ... * 91^(2^n) (mod 75)

3. 
   Biz dastlabki qadamda 91^1 ni hisobladik, keyingisida esa undan olingan darajalarini yig'indilari bilan yana o'zingizni e'lon qilishingiz kerak.
   

91^1 (mod 75) = 16.

4. 
   Keyingi daraja 2 darajasi bo'ladi:
   

91^2 (mod 75) = (91^1 * 91^1) (mod 75) = (16 * 16) (mod 75) =256 (75 ga qoldiqsiz), bu esa 1 ga teng.

5. 
   91^4 darajani hisoblaymiz:
   

91^4 (mod 75) = (91^2 * 91^2) (mod 75) = (1 * 1) (mod 75) = 1.

6. 
   Endi 91^(2^3) ni hisoblaymiz:
   

91^8 (mod 75) = (91^4 * 91^4) (mod 75) = (1 * 1) (mod 75) = 1.
Bu jarayonni davom ettirishimiz kerak. Darajani darajaga oshirib, 91^(2^4), 91^(2^5), va h.k. darajalarni hisoblashimiz kerak.
So'ngra natijalarni yig'indisi orqali 91^746 (mod 75) ni hisoblaymiz:

91^746 (mod 75) = 91^(2^0) * 91^(2^1) * 91^(2^2) * 91^(2^3) * ... * 91^(2^n) (mod 75)

91^746 (mod 75) = 16 * 1 * 1 * 1 * ... * 1 (mod 75)

Chunki barcha darajalar 1 ga teng, natija:

91^746 (mod 75) = 16 * 1 * 1 * 1 * ... * 1 (mod 75) = 16 (mod 75)

Demak, 91^746 (mod 75) = 16 ga teng.