The number of integers $$n$$ such that $$1 \leq n \leq 10^7$$ and $$\gcd(n, 10^7) = 10^3$$ is ____

$$\gcd(n, 10^7) = 10^3$$ forces $$n$$ to have *exactly* three factors each of 2 and 5. Write $$n = 10^3 \cdot k$$; then $$\gcd(k, 10^4) = 1$$ (i.e., $$k$$ shares no factor of 2 or 5 with $$10^4$$).

Range: $$1 \leq n \leq 10^7 \Rightarrow 1 \leq k \leq 10^4$$.

Count = number of integers in $$[1, 10^4]$$ coprime to $$10^4 = 2^4 \cdot 5^4 = \varphi(10^4)$$.

$$\varphi(10^4) = 10^4 \left(1 - \tfrac{1}{2}\right)\left(1 - \tfrac{1}{5}\right) = 10^4 \cdot \tfrac{1}{2} \cdot \tfrac{4}{5} = \mathbf{4000}$$.