Answer:
The smallest composite number for [tex]n[/tex] is 24, which is the product 2 and 12.
Step-by-step 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 [tex]n[/tex] the smallest composite number such that there are two positive integers [tex]x[/tex], [tex]y[/tex] such that [tex]x - y = 10[/tex] and [tex]x\cdot y = n[/tex]. [tex]n[/tex] is a positive integer, since both integers are positive. Then, we can use the following formula:
[tex](y+10)\cdot y = n[/tex]
[tex]y^{2}+10\cdot y = n[/tex]
[tex]y^{2}+10\cdot y - n = 0[/tex] (1)
We obtain the roots of this second order polynomial by means of the Quadratic Formula:
[tex]y = \frac{-10\pm \sqrt{10^{2}-4\cdot (1)\cdot (-n)}}{2}[/tex]
[tex]y = -5\pm \sqrt{25+n}[/tex] (2)
Given that [tex]y > 0[/tex], we must observe the following inequation:
[tex]-5\pm \sqrt{25+n}>0[/tex] (3)
[tex]\sqrt{25+n} > 5[/tex]
[tex]25 + n > 25[/tex]
[tex]n > 0[/tex]
Let is find the values of x and y by iterative means:
x = 11, y = 1
[tex](11)\cdot (1) = 11[/tex]
11 is not a composite number, but a prime number.
x = 12, y = 2
[tex](12)\cdot (2) = 24[/tex]
24 is a composite number, which contains the following product of prime numbers:
[tex]2^{3}\cdot 3 = 24[/tex]
The smallest composite number for [tex]n[/tex] is 24, which is the product 2 and 12.
