Factoring a large integer

Small primes. $1001$ is odd, digit sum $2$, not divisible by $3$, ends in $1$, not divisible by $5$.

Try $7$. $1001=7\cdot143$.

Factor $143$. $143=11\cdot13$.

Conclusion. $1001=7\cdot11\cdot13$.

Alternative. $1001=10^3+1=(10+1)(10^2-10+1)=11\cdot91=11\cdot7\cdot13$.