ATRACTOR

Let us consider another question:

In what follows, whenever we mention a smalllest divisor we mean divisors different from \(1\).

We start, again, with some concrete examples.

What are the numbers whose smallest divisor is \(2\)? And what are the numbers whose smallest divisor is \(4\)? And what are the numbers whose smallest divisor is \(6\)?

What are the numbers whose smallest divisor is \(5\)? And what are the numbers whose smallest divisor is \(7\)?

Use the preceding applet to answer the questions above. Try to reach a more general conclusion... before moving on.

When looking for an answer to the above questions, you might have concluded that the numbers whose smallest divisor is \(2\) are precisely all even numbers. On the other hand, there is no number of which \(4\) is the smallest divisor, since, as already explained above, all numbers divisible by \(4\) are also divisible by \(2\), so the multiples of \(4\) have \(2\) as their smallest divisor. Likewise, there is also no number of which \(6\) is the smallest divisor, since all numbers divisible by \(6\) are also divisible by \(2\) and by \(3\).

The first number, distinct from \(5\), for which \(5\) is the smallest divisor is \(5\times 5=25\). And, the first number, distinct from \(7\), for which \(7\) is the smallest divisor is \(7\times 7=49\).