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

Need $\gcd(n, 10^7) = 10^3$, with $1 \leq n \leq 10^7$.

Since $10^3$ divides $n$, write $n = 10^3 \cdot m$ for some positive integer $m$.

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From $n \leq 10^7$:

$m \leq 10^4$

Also, $\gcd(10^3 \cdot m,\, 10^7) = 10^3 \cdot \gcd(m, 10^4)$.

For this to equal $10^3$, we need:

$\gcd(m, 10^4) = 1$

Since $10^4 = 2^4 \cdot 5^4$, this just means $m$ is coprime to $10$.

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Count integers $m$ in $[1, 10^4]$ coprime to $10$. The fraction of such integers is:

$\left(1 - \dfrac{1}{2}\right)\left(1 - \dfrac{1}{5}\right) = \dfrac{1}{2} \cdot \dfrac{4}{5} = \dfrac{2}{5}$

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Required count $= 10^4 \cdot \dfrac{2}{5} = 4000$

Answer $= 4000$