Question

Gasiti cel mai mic numar n€N* Astfel incat 240n sa fie
a)patrat perfect
b)cub perfect

Answer 1

 [tex]\it a)\ \ A = 240n =2^4\cdot3\cdot5n =16\cdot15n = 4^2\cdot15n \\\;\\ A\ este\ p\breve{a}trat \ perfect\ pentru \ 15n = \ p\breve{a}trat \ perfect \Rightarrow n = 15[/tex]

[tex]\it\ b)\ A = 240n = 8\cdot30n =2^3\cdot30n \\\;\\ A\ este\ cub\ perfect\ pentru \ 30n = cub\ perfect \Rightarrow n = 30^2 = 900[/tex]

Answer 2

a)...  240×n = 24×10×n = 8×3×2×5×n=(2²)²×3×5×n ⇒ n = 3×5 = 15
            adica: 240×15=3600=60² = p.p.

b)...  240×n = (2²)²×3×5×n ⇒ n = 2²×3²×5² = 900
           adica: 240×450 = 2⁴×3×5 × 2²×3²×5² = 2⁶×3³×5³ = [(2²)×3×5]³ = c.p.