Vastaus:
Loput ovat #=0#
Selitys:
Suorita tämä aritmeettisella kongruenssimoduulilla #7#
#"ensimmäinen osa"#
#111 67#
#333 18 47#
#4^2 27#
#4^3 17#
Siksi, #333^444 4^4447 (4^3)^148 1^148 17#
#"toinen osa"#
#111 67#
#444 24 37#
#3^2 27#
#3^3 -17#
Siksi, #444^333 (3)^3337 ((3)^111)^3 (-1)^3 -17#
Lopuksi
#333^444+444^333 1-1 07#
Vastaus:
# 333 ^ 444 + 444 ^ 333 = 0 (mod 7) #
Selitys:
# 333 = 4 (mod 7) #
# 333 ^ 2 = 4 ^ 2 = 2 (mod 7) #
# 333 ^ 3 = 4 ^ 3 = 1 (mod 7) #
Johdosta # 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) #
Johdosta # 333 = 3 (Mod 6) #, # 444 ^ 333 = 3 ^ 3 = 6 (mod 7) #
Täten, # 333 ^ 444 + 444 ^ 333 = 1 + 6 = 7 = 0 (mod 7) #