Question

Three consecutive positive integers are raised to the first, second and third powers respectively and then added The sum so obtained is a perfect square whose square root equals the total of the three original integers Which of the following best describes the minimum say m of these three integers?

• A. $1\le m\le 3$
• B. $4\le m\le 6$
• C. $7\le m\le 9$
• D. $10\le m\le 12$

Answer 1

The correct option is A $1\le m\le 3$

Let the three integers be $m-1,m,m+1$
As given,
$m-1+m^2+(m+1{)}^3=(m-1+m+m+1{)}^2$
$m-1+m^2+m^3+1+3m^2+3m=9m^2$
$m^3-5m^2+4m=0$
$m(m^2-5m+4)=0$
$m(m-1)(m-4)=0$
$m=0,1,4$
$m$ cannot be 0 o 1 else the smallest number will be either -1 or 0 which is not possible.
Hence, $m=4$
and smallest number then is $3$
Thus correct answer is $1\le m\le 3$