Koji je ostatak od 333 ^ 444 + 444 ^ 333 podijeljen sa 7?

Odgovor:

Ostatak je #=0#

Izvedite to pomoću aritmetičke kongruencije po modulu #7#

#"prvi dio"#

#111 67#

#333 18 47#

#4^2 27#

#4^3 17#

Stoga, #333^444 4^4447 (4^3)^148 1^148 17#

#"drugi dio"#

#111 67#

#444 24 37#

#3^2 27#

#3^3 -17#

Stoga, #444^333 (3)^3337 ((3)^111)^3 (-1)^3 -17#

Konačno, #333^444+444^333 1-1 07#

# 333 ^ 444 + 444 ^ 333 = 0 (Mod 7) #

# 333 = 4 (Mod 7) #

# 333 ^ 2 = 4 ^ 2 = 2 (Mod 7) #

# 333 ^ 3 = 4 ^ 3 = 1 (Mod 7) #

Zbog # 444 = 0 (Mod 3) #, # 333 ^ 444 = 3 ^ 0 = 1 (Mod 7) #

# 444 = 3 (Mod 7) #

# 444 ^ 2 = 3 ^ 2 = 2 (Mod 7) #

# 444 ^ 3 = 3 ^ 3 = 6 (Mod 7) #

# 444 ^ 4 = 3 ^ 4 = 4 (Mod 7) #

# 444 ^ 5 = 3 ^ 5 = 5 (Mod 7) #

# 444 ^ 4 = 3 ^ 6 = 1 (Mod 7) #

Zbog # 333 = 3 (Mod 6) #, # 444 ^ 333 = 3 ^ 3 = 6 (Mod 7) #

Tako, # 333 ^ 444 + 444 ^ 333 = 1 + 6 = 7 = 0 (Mod 7) #