18. a) abcd = 100ab + cd = 100ab +2ab = 102ab = 17·6·ab ÷ 17 = 6·ab r=0
b ) 1000a + bcd =(a+1)·bcd + a+ 2
1000a + bcd = a·bcd + bcd + a + 2
999·a = a·bcd + 2 a(999 -bcd) =2 a = 1 999 - bcd = 2 bcd = 997
abcd = 1997 a = 2 999- bcd = 1 bcd = 998 abcd = 2998
1997 = 997 ·2 + 3 2998 = 998·3 + 4
c) abc +
bc
c
-----------------
349 ⇒ c = 3 b = 2 a = 3
20. a)
a1 = n·c1 +r1
a2 = n·c2 + r2
-------------------
a10 = n·c10 + r10
exista 10 caturi (diferite) si 10 resturi diferite r < n r ∈{0,1,2,....(n-1)}
daca, r1 = 0 r10 = 9 n-1 = 9 n≥10
b) S = a1 + a2 + ....+a10 = (10·1 + 0) + (10·1+1) + (10·1+2) +......+(10·1 +9)
S = 10·10 + (1 + 2 +3 +.....+ 9) = 100 + 45= 145
