Question

SA SE ARATE CA NUMARUL
M=1997+2×(1+2+.....+1996)ESTE PATRAT PERFECT
M=111+222LA PUTEREA 2+333LA PUTEREA3+444LA PUTEREA 4+555LAPUTEREA5-12345NU ESTE PATRAT PERFECT

Answer 1

1)M=1997+2·(1+2+3+...+1996)
M=1997+2·(1996·1997):2
M=1997+1996·1997
Dam factor comun 1997:
M=1997·(1+1996)
M=1997·1997
M=1997²

1) Un nr nu este patrat perfect daca are ultimele cifre 2,3,7 sau 8. Operam astfel:
U(222^2)=U(2^2)=4
U(333^3)=U(3^3)=7
U(444^4)=U(4^4)=6
U(555^5)=U(5^5)=5
Calculam: 1+4+7+6+5-5=5+7+6+5-5=12+6+5-5=18+5-5=23-5=18. Ultima cifra=8⇒M≠ patrat perfect