Gọi $3$ số nguyên liên tiếp đó là : $n$ ; $n$ + $1$; $n$ + $2$ ( $n$ ∈ $Z$ , $n$ $\neq$ $0$ )( * )
Theo bài ra , ta có :
$n$ . ( $n$ + $1$ ) + $n$ . ( $n$ + $2$ ) + ( $n$ + $1$ ) . ( $n$ + $2$ ) = $242$
⇔ ( $n$² + $n$ ) + ( $n$² + $2$$n$ ) + ( $n$² + $3$$n$ + 2 ) = $242$
⇔ $3$$n$² + $6$$n$ + $2$ = $242$
⇔ $3$$n$² + $6$$n$ = $242$ - $2$ = $240$
⇔ $3$$n$² + $6$$n$ - $240$ = $0$
⇔ $3$ . ( $n$² + $2$$n$ - $80$ ) = $0$
⇔ $n$² + $2$$n$ - $80$ = $0$
⇔ $n$² - $8$$n$ + $10$$n$ - $80$ = $0$
⇔ $n$ . ( $n$ - $8$ ) + $10$ . ( $n$ - $8$ ) = $0$
⇔ ( $n$ - $8$ ) . ( $n$ + $10$ ) = $0$
⇔ $n$ - $8$ = $0$ hoặc $n$ + $10$ = $0$
⇔ $n$ = $8$ hoặc $n$ = $ - 10$
⇔ $n$ + $1$ = $9$ , $n$ + $2$ = $10$ hoặc $n$ + $1$ = $ - 9$ , $n$ + $2$ = $ - 8 $ ( thỏa mãn điều kiện ( * ) )
Vậy $3$ số nguyên liên tiếp đó thỏa mãn đề bài là :
⇒ ( $n$ , $n$ + $1$ , $n$ + $2$ ) ∈ { ± $8$ ; ± $9$ ; ± $10$ }
