Strategic Way of Factoring Modulus ``$N = p^r q$

Cryptography is fundamental to the provision of a wider notion of information security. Electronic information can easily be transmitted and stored in relatively insecure environments. This research was present to factor the prime power modulus $N = p^r q$ for $r \geq 2$ using the RSA key equation, if $\frac{y}{x}$ is a convergents of the continued fractions expansions of $\frac{e}{N - \left(2^{\frac{2r+1}{r+1}} N^{\frac{r}{r+1}} - 2^{\frac{r-1}{r+1}} N^{\frac{r-1}{r+1}}\right)}$. We furthered our analysis on $n$ prime power moduli $N_i = p_i^r q_i$ by transforming the generalized key equations into Simultaneous Diophantine approximations and using the LLL algorithm on $n$ prime power public keys $(N_i,e_i)$ we were able to factorize the $n$ prime power moduli $N_i = p_i^r q_i$, for $i = 1,....,n$ simultaneously in polynomial time.