Past exam of the analytic number theory and elliptic curves course of the University of Oxford / 2007 / B / 8 / i / Solution

The doubling slope at $P=(2,1)$ is $(3\cdot 2^2-1)/(2\cdot 1)=11/2$, givingThese denominators are invertible modulo $N$. To add $P$ again, the difference of first coordinates is $105/4-2=97/4$, so the new slope would be $-1083/194$. This requires inversion of a number divisible by 97. The elliptic curve method instead computesIt is a proper factor. The failure of the group-law division modulo the composite integer reveals different behavior modulo its prime factors.