# $S$-prime numbers Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

## Suggestion

Let $S$ be the set of all positive integers that are $1$ more than a multiple of $10$, so $S = \{1, 11, 21, 31, 41, \dotsc\}$.

We say that an element $x$ of the set $S$ is $S$-prime if $x > 1$ and whenever the elements $a$ and $b$ of the set $S$ satisfy $ab = x$ we have $a = 1$ or $b = 1$.

Are there distinct $S$-prime numbers $a$, $b$, $c$ and $d$ such that $ab = cd$?

We’ve been given a new definition (of $S$-prime numbers), so it would be a good idea to try to understand that properly before trying to tackle the question. Can you come up with some examples of numbers that are $S$-prime, and also some examples of numbers that are not $S$-prime?

For example, $21$ is $S$-prime, because if $ab = 21$ and $a$ and $b$ are both $1$ more than a multiple of $10$, then $a = 1$ or $b = 1$. But $121$ is not $S$-prime, because $121 = 11 \times 11$ and $11$ is a member of $S$ but does not equal $1$.