Modular Arithmetic in RSA Consider the following the following RSA public key $pk = (N, e) = (1457, 1307)$.

(a) Knowing that $187^2 \equiv 1 \pmod {1457}$ find the factorization of $N$.

(b) Given the factorization of $N$ computed above, use the CRT to decrypt the following ciphertext
$c = E_{pk}(m) = 3$.

For part a) I am not sure which property of modular arithmetic can be used? I know $187$  has a multiplicative inverse $187$. I also know the Euler function. But I think I am missing how to calculate $\phi$ from this information.
For part a) of your question: In this case 2 would be the order of 187 in the multiplicative group modulo 1457. This means you could retrieve the factors of 1457 using the formula gcd($187^{2/2}-1$, 1457) and gcd($187^{2/2}+1$, 1457). 

Having the factors of 1457, you can compute $\phi$ with ease.

@fgrieu: The calculation is taken from the non-quantum part of Shor's Algorithm. Basically it's the same as what you mentioned in the comments of the question. Thanks for pointing out my mistake. I mixed up some numbers in my gcd calculation...