Gọi 4 số liên tiếp cần tìm là : \(a;a+1;a+2;a+3\left(a\in N\right)\)
Theo bài ra ta có :
\(a\left(a+1\right)\left(a+2\right)\left(a+3\right)=24\)
\(\Leftrightarrow\left[a\left(a+3\right)\right]\left(a+1\right)\left(a+2\right)=24\)
\(\Leftrightarrow\left(a^2+3a\right)\left(a^2+3a+2\right)=24\)
\(\Leftrightarrow\left(a^2+3a\right)^2+2\left(a^2+3a\right)=24\)
\(\Leftrightarrow\left(a^2+3a\right)^2+2\left(a^2+3a\right)+1=25\)
\(\Leftrightarrow\left(a^2+3a+1\right)^2=25\)
Do \(a\in N\Rightarrow a^2+3a+1>0\forall a\)
\(\Rightarrow a^2+3a+1=5\)
\(\Rightarrow a^2+3a-4=0\)
\(\Rightarrow a^2-a+4a-4=0\)
\(\Rightarrow a\left(a-1\right)+4\left(a-1\right)=0\)
\(\Rightarrow\left(a+4\right)\left(a-1\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}a-1=0\\a+4=0\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}a=1\\a=-4\left(L,a\in N\right)\end{matrix}\right.\)
\(\Rightarrow a=1;a+1=2;a+2=3;a+3=4\)
Vậy 4 số tự nhiên liên tiếp là : \(1,2,3,4\)
