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Question Number 207320 by BaliramKumar last updated on 11/May/24

Commented by A5T last updated on 11/May/24

19. The remainder when a number is divided by 16 is the same as the remainder when its last four digits are divided by 16 9100≡12(mod 16) ⇒(b)

19.Theremainderwhenanumberisdividedby16isthesameastheremainderwhenitslastfourdigitsaredividedby169100≡12(mod16)⇒(b)

Answered by A5T last updated on 11/May/24

18. 6^n ≡^(10) 6;11^n ≡^(10) 1; 9^n ≡^(10) 1 when n=2k;9^n ≡^(10) 9 when n=2k+1 4^n ≡^(10) (−6)^n ≡^(10) 6 when n=2k; 4^n ≡^(10) 4 when n=2k+1 when n=2k: 4^n +6^n +9^n +11^n ≡6+6+1+1≡^(10) 4 n=2k+1: 4^n +6^n +9^n +11^n ≡4+6+9+1≡^(10) 0 ⇒Sum of distinct remainders=0+4=4⇒(b)

18.6n≡106;11n≡101;9n≡101whenn=2k;9n≡109whenn=2k+14n≡10(−6)n≡106whenn=2k;4n≡104whenn=2k+1whenn=2k:4n+6n+9n+11n≡6+6+1+1≡104n=2k+1:4n+6n+9n+11n≡4+6+9+1≡100⇒Sumofdistinctremainders=0+4=4⇒(b)

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