Question

Let $n$ be the smallest composite number such that it can be written as the product of two positive integers that differ by $10.$ How many distinct prime factors does $n$ have?

Answer 1

Compute the number of prime factors.

Consider the positive integer as $x$ and $y$.

According to the given condition

$$\begin{gathered} y-x=10 \\ \therefore y=x+10 \end{gathered}$$

Since $n$ is the product of the positive integers ,use the first positive integers values for $x$ to find the composite number $n$.

At $x=0$ ,we have

$$\begin{gathered} n=0(10+0) \\ \therefore n=0 \end{gathered}$$.

$0$ is not a composite number.

At $x=1$,

we have $n=1(1+10)=11$..... which is again not a composite number.

At $x=2$,

we have $n=2(2+10)=24$......which is a composite number.

Therefore. $n=24$is the smallest composite number

Now, factors of $24$ are $24=2\times 2\times 2\times 3$.

Hence, the distinct prime factors are $2$ and $3$.