## 1640.61 – The Largest What! The Smallest What!

What is the largest integer less than 9,000,000 whose smallest prime divisor is 3163?

Solution

Let's start by taking a look at 3163 itself. First of all, 3163 IS an integer less than 9,000,000. We'll see about its being the largest such in a minute. But is the smallest prime divisor of 3163 3163 itself? Sure it is, because 3163 is prime and thus is the ONLY prime divisor of 3163.

But is 3163 the LARGEST integer less than 9,000,000 whose smallest prime divisor is 3163?

Suppose that $n$ is a larger integer with 3163 as its smallest prime divisor. Then $n$ factors, $n = 3163x$, where $x \ge 3163$ since all the prime factors of $n$ have to be greater than (or equal to) 3163. This means that

$n = 3163 x \ge 3163 \cdot 3163 = 10,004,569,$

so $n$ is bigger than 9,000,000.

It follows that 3163 is the ONLY integer less than 9,000,000 whose smallest prime divisor is 3163. That makes 3163 also the LARGEST integer less than 9,000,000 whose smallest prime divisor is 3163.