" a, b, c, d " Z \ {0}, " n " N
a|b Ð!Ò! (ac)|(bc) 120 | n5 - 5n3 + 4n a|b '" b|a Ð!Ò! a = Ä…b
2|a(a + 1) a|b '" b|c =Ò! a|c a|b =Ò! " c " Z a|(bc)
a|0 '" 1|a '" a|a n = 1 =Ò! 2n 3n + 1 3|a2 + b2 =Ò! 3|a '" 3|b

4 (a2 Ä… 2) a|b '" a|c =Ò! a2|(bc) a|b '" c|d =Ò! (ac)|(bd)
a|b =Ò! |a| | |b| 2 (n3 - (n + 1)3)
a|1 Ð!Ò! a = Ä…1 2 a '" 3 a =Ò! 24|(a2 - 1)
a|b =Ò! |a| d" |b| d|(10b - 1) '" d|(10a + c) =Ò! d|(a + bc)
3|a(a + 1)(a + 2) a|b Ð!Ò! a| - b Ð!Ò! -a|b Ð!Ò! -a| - b
a, b, c, d " Z a2 + b2 + c2 = d2
a, c " Z, n " N a c n
a - c = nk k
4k + 1 4k + 3
k
3k 3k + 1
k
9k, 9k + 1, 9k + 8 k
a a2 " 8Z + 1.
b " Z \ {Ä…1} 2a2 Ä… 1, a " Z (ab)2
10n + 1
t
W = a0 + a1x + ... + an-1xn-1 + xn t " Z
" a, b, c " Z \ {0} a|(b + c) =Ò! a|b (" a|c
5|a2 + b2 + c2 Ð!Ò! 5|a '" 5|b '" 5|c
d|(a + bc) =Ò! d|(10b - 1) '" d|(10a + c)
 a, b, c " Z a|c '" b|c =Ò! ab|c
a b
a = Ä…79, b = Ä…11 a = Ä…6, b = Ä…14
a2 + b2 = 2ab + (a - b)2 (a + b)a + (a + b)b = (a + b)2
p
p " P " k " N k < p =Ò! p |
k