Question

The product of three successive natural numbers, the middle number being the square of a natural number, is divisible by $60$

• A. True
• B. False

Answer 1

The correct option is A True

Since the middle number of the three consecutive numbers is x, the other two numbers are $(x-1)$ and $(x+1).$

According to the given condition, we have

$x^2=\left[\right(x+1{)}^2-(x-1{)}^2]+60$

$x^2=(x^2+2x+1)-(x^2-2x+1)+60=4x+60$

$x^2-4x-60=0=>(x-10)(x+6)=0$

$x=10$ or $-6$

Since $x$ is a natural number, we get $x=10$

Hence the three numbers are $9,10,11$