Question

Find the smallest of three consecutive positive integers if the product of the smaller two

integers is 2 more than 8 times the largest integer.

Answer 1

9

Explanation:

Let the three consecutive positive integers be:

• x
• x + 1
• x + 2

If the product of the two smaller integers is 2 more than 8 times the largest integer then:

`\implies x(x+1)=8(x+2)+2`

Solve the equation for x:

`\implies x^2+x=8x+16+2`

`\implies x^2+x=8x+18`

`\implies x^2-7x-18=0`

`\implies x^2-9x+2x-18=0`

`\implies x(x-9)+2(x-9)=0`

`\implies (x+2)(x-9)=0`

Therefore:

`\implies x+2=0 \implies x=-2`

`\implies x-9=0 \implies x=9`

As x is a positive integer, x = 9.