a) Aflati toate numerele de forma 23ab care se impart cu rest 0 la 42
23ab:42=x rest 0
cel mai mic: 2300:42=54 rest 32 => x>54
cel mai mare2399:42=57 rest 5 => x≤57
=> 54<x≤57
Deci 23ab∈ {55*42, 56*42, 57*42}
23ab ∈ {2310, 2352, 2394}
b) Aflati toate numerele de forma 51ab care se impart cu rest 0 la 21
51ab : 21 =x, rest 0
cel mai mic numar de forma 51ab este: 5100:21=242 rest 18 => x>242 ,
cel mai mare numar de forma 51ab este: 5199:21=247 rest 12 => x≤ 247 ,
Deci 242< x ≤247 =>
=> 51ab ∈ {243*21, 244*21,245*21,246*21,247*21}
Adica 51ab ∈{5103, 5124, 5145, 5166, 5187}
c)Aflati numerele naturale care impartite la 37 dau restul egal cu catul
X: 37 =C restC , stim ca restul <37
X=37C+C=38C
daca C=0, x=37 C=1 => x= 38
C=2 => x=38*2=76
C=3 => x=38*3=114
.....
C=36 => x=38*36=1368
Deci x∈ {37,38,76, 114, 152, 190, 228, 266, 304, 342, 380, 418, 456, 494, 532, 570, 608, 646, 684, 722, 760, 798, 836, 874, 912, 950, 988, 1026, 1064, 1102, 1140, 1178, 1216, 1254, 1292, 1330, 1368}
d)Calculati suma
numerelor naturale care impartite la 8 dau catul 13
Restul <8
8*13+0=104
8*13+1=105
8*13+2=106
8*13+3=107
8*13+4=108
8*13+5=109
8*13+6=110
8*13+7=111
suma: 104+105+106+107+108+109+110+111= (111+104)*8:2=215*4=860
