Sieve of Eratosthenes

Question II

Let us consider another question:

2. Of which numbers is a certain given (non-negative) number the smallest divisor (different from 1)?

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 $$7\times 7=49$$.