Let x be a positive real number and let d be a positive integer. Prove that the number of positive integers less than or equal to x that are divisible by d is [x/d].
I am having trouble with this one. If anyone could help that would be terrific!
Let x be a positive real number and let d be a positive integer. Prove that the number of positive integers less than or equal to x that are divisible by d is [x/d].
I am having trouble with this one. If anyone could help that would be terrific!
Write $\displaystyle \frac{x}{d}=a_0+r\,,\,\,a_0\in\mathbb{N}\,,\,\,0\l eq r < 1$ $\displaystyle \Longrightarrow \left[\frac{x}{d}\right]=a_0$
But $\displaystyle a_0$ is the number of whole (integer) times that d "fits" into x, meaning: $\displaystyle d, 2d, 3d,\ldots,a_0d$ are all multiples of d less than or equal x, and every multiple of d less than or equal x must be one of these (why?), so...
Tonio