Question

The recursive definition of the set S of positive integers that are multiples of 5 is

Answer 1

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Answer 2

Consider letting \(x = 5n\) and \(y=5m\), then ensuring all of them would still be multiples of \(5\).

Answer 3

For example \[
x+y = 5n+5m = 5(m+n)
\]So it's still multiple of \(5\).

The next step is to figure out whether we cover all multiples of \(5\).

Answer 4

For example, consider the second option: \[
5y = 5(5n) = 5^2n
\]We will really only get numbers of the form \(5^kn\).
When we say \(5\in S\) it let's us know that \(n=1\) to start with. We can't get \(10 = 2\times 5\) because there is no \(k\) where \(5^kn = 10\).

So the second option won't work because it doesn't include all multiples of 5.

Answer 5

Thank you, Much appreciated!