What is the remainder when 57^54 is divided by 176

Question

What is the remainder when 57^54 is divided by 176?

BG · Answer

This Q can be split into (19*3)^54/11*16 or (19^54)*(3^54)/11*16 which is equal to (3^54/11)*(19^54/16).
 Remainders are as follows:
 for (3^54/11) rem is 4. 3^5/11=243/11 has rem 1. 3^54={ *3^4}/11 and rem is 4

FOr (19^54/16) rem is 9
19^54=(19^4)^13 where (19^4)/16 has remainder 1 
Now rem for ( 57^54/176) is 4*9=36
Rem is IMO 36

yezz · Answer

GMATT73 WUT IS OA PLZ

haas_mba07 · Answer

ok.. this took looooonng... but have the satisfaction of getting this. :-)

So when 57 and its powers are divided by 176, the cyclicity .i.e. the remainders repetition is at 10. i.e the remainders start repeating at 10. Anyway, the remainders are (% is modulo sign): 57 %176 = 57 57^2%176 = 81 57^3%176 = 41 57^4%176 = 49 57^5%176 = 153 57^6%176 = 97 57^7%176 = 73 57^8%176 = 113 57^9%176 = 105 57^10%176 = 1 ---------------------------- Repeats .. .. .. 57^50%176 = 1 57^51%176 = 57 57^52%176 = 81 57^53 %176 = 41 57^54%176 = 49 Remainder : 49. I am sure someone has a much-much more elegant and smarter way of doing this...

yezz · Answer

OA plz and OE if possible

What is the remainder when 57^54 is divided by 176

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