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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?

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Solution

## Compute the number of prime factors.Consider the positive integer as $x$ and $y$.According to the given condition $y-x=10\phantom{\rule{0ex}{0ex}}\therefore y=x+10$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 $n=0\left(10+0\right)\phantom{\rule{0ex}{0ex}}\therefore n=0$.$0$ is not a composite number.At $x=1$,we have $n=1\left(1+10\right)=11$..... which is again not a composite number.At $x=2$,we have $n=2\left(2+10\right)=24$......which is a composite number.Therefore. $n=24$is the smallest composite numberNow, factors of $24$ are $24=2×2×2×3$.Hence, the distinct prime factors are $2$ and $3$.

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