Question

cho a b c nguyên dương tìm a b c sao cho a³+3a²+5=5b² và  a+b=5^c

Answer 1

Do \(a;b;c\in Z^+\Rightarrow5^b=a^3+3a^2+5>a+3=5^c\)

\(\Rightarrow5^b>5^c\Rightarrow b>c\)

\(a^3+3a^2+5=5^b\)

\(\Rightarrow a^2\left(a+3\right)+5=5^b\)

\(\Rightarrow a^2\cdot5^c+5=5^b\)

\(\Rightarrow5^b⋮5^c\)

\(\Rightarrow a^2\left(a+3\right)+5⋮a+3\)

\(\Rightarrow5⋮a+3\)

\(\Rightarrow a+3\in\left\{5,1,-1,-5\right\}\)

Mà \(a+b>3\Rightarrow a+3=5\)

\(\Rightarrow a=2\)

\(\Rightarrow b=2;c=1\)