Determine the order of the following integers modulo 2520:
$$ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \pmod {2520} $$
The prime factorisation of 2520 is $2^3 \times 3^2 \times 5 \times 7$.
This means the $\gcd(n,2520) \ne 1$ for $n=2,3,4,5,6.7.8.9.10$, and so the order does not exist for all the given integers except 1.
The only order that exists is the order of 1 modulo 2520 which is 1.