Question

1)Aratati ca nr. a=2003+2.(1+2+......+2002)este patrat perfect
                  
2)aratati ca nr. b=1+3+5...+2011 este patrat perfect
3)aratati ca nr. a=81+2.81+...+49  .  81 este patrat perfect

Answer 1

[tex]a=2003+2*(1+2+......+2002) \\ a=2003+2 \frac{2002(2002+1)}{2} \\ a=2003+2002(2002+1) \\ a=2003+2002*2003 \\ Dam~factor~comun~pe~2003: \\ a=2003(1+2002) \\ a=2003*2003 \\ a=2003^2=p.p[/tex]

[tex]b=1+3+5...+2011 \\ b= \frac{670(2011+1)}{2} \\ b= \frac{1006*2012}{2} \\ b=1006*1006 \\ b=1006^2=p.p[/tex]

[tex]a=81+2*81+...+49*81 \\ Dam~factor~comun~pe~81: \\ a=81(1+2+3+....+49) \\ a=81* \frac{49(49+1)}{2} \\ a=81* \frac{49*50}{2} \\ a=81*49*25 \\ a=9^2*7^2*5^2 \\ a=(9*7*5)^2=p.p[/tex]