Need help in Apostol Calculus proof

[zjhok2004]

let b denote a fixed positive integer. Prove the following statement by induction: for every integer n≥0, there exist nonnegative integers q and r such that n= qb+r, 0≤r<b.



Can someone help me on how to solve this question? and how does induction works here?
thank you

 

[voko]

Are there any restrictions on q and b? Otherwise the statement is trivially satisfied by q = n, b = 1, r = 0.

 

[zjhok2004]

Are there any restrictions on q and b? Otherwise the statement is trivially satisfied by q = n, b = 1, r = 0.

I think you have to prove by using induction

 

[jbunniii]

Are there any restrictions on q and b? Otherwise the statement is trivially satisfied by q = n, b = 1, r = 0.

I don't think you get to choose b.

 

[jbunniii]

Can someone help me on how to solve this question? and how does induction works here?
thank you

You can easily check that it's true for $n = 0$. Now suppose it's true for $n$, so there exist $q$ and $r < b$ such that $n = qb + r$. Now consider $n + 1$. A reasonable first step would be to add 1 to both sides of the equation above:
$$n + 1 = qb + r + 1$$
What does this do for you?