Question

Let $n$ be the smallest composite number such that it can be written as the product of two positive integers that differ by 10?

Answer 1

The smallest composite number for
`n` is 24, which is the product 2 and 12.

Explanation:

A composite number is an integer that is not a prime number, that is, a number that can be divided only by one and itself. Let be
`n` the smallest composite number such that there are two positive integers
`x`,
`y` such that
`x - y = 10` and
`x\cdot y = n`.
`n` is a positive integer, since both integers are positive. Then, we can use the following formula:

`(y+10)\cdot y = n`

`y^(2)+10\cdot y = n`

`y^(2)+10\cdot y - n = 0` (1)

We obtain the roots of this second order polynomial by means of the Quadratic Formula:

`y = \frac{-10\pm \sqrt{10^(2)-4\cdot (1)\cdot (-n)}}{2}`

`y = -5\pm √(25+n)` (2)

Given that
`y > 0`, we must observe the following inequation:

`-5\pm √(25+n)>0` (3)

`√(25+n) > 5`

`25 + n > 25`

`n > 0`

Let is find the values of x and y by iterative means:

x = 11, y = 1

`(11)\cdot (1) = 11`

11 is not a composite number, but a prime number.

x = 12, y = 2

`(12)\cdot (2) = 24`

24 is a composite number, which contains the following product of prime numbers:

`2^(3)\cdot 3 = 24`

The smallest composite number for
`n` is 24, which is the product 2 and 12.