grandes-ecoles 2024 Q21a

grandes-ecoles · France · x-ens-maths-a__mp Number Theory Combinatorial Number Theory and Counting
Let $x$ be a positive real number greater than or equal to 1 and $q \in \mathbb{N}^*$. Justify that the quantity $$\operatorname{Card}\{n \in \mathbb{N} \cap [1,x] : n \equiv 0 \pmod{q}\} - \frac{x}{q}$$ is bounded in absolute value by a real number independent of $x$ and $q$.
Let $x$ be a positive real number greater than or equal to 1 and $q \in \mathbb{N}^*$. Justify that the quantity
$$\operatorname{Card}\{n \in \mathbb{N} \cap [1,x] : n \equiv 0 \pmod{q}\} - \frac{x}{q}$$
is bounded in absolute value by a real number independent of $x$ and $q$.