Patrat perfect cu 2n cifre (O.J.)

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Marcelina Popa
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Patrat perfect cu 2n cifre (O.J.)

Post by Marcelina Popa »

Fie numarul \( A=111...1 \), in care cifra \( 1 \) apare de \( 2n \) ori si \( B=222...2 \), in care cifra \( 2 \) apare de \( n \) ori. Aratati ca \( A-B \) este patrat perfect.
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Claudiu Mindrila
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Post by Claudiu Mindrila »

\( \underbrace{11..1}_{2n}- \underbrace{22...2}_{n}=\underbrace{11...1}_{n}(10^n+1-2)=\frac{10^n-1}{9}\cdot (10^n-1)=\left( \frac{10^n-1}{3} \right)^2=\underbrace{33...3}_{n}^2 \)
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