= Solution
{c}
In the <elliptic curve method>, addition is attempted modulo the composite number. A denominator that is not a <unit in a ring> supplies a factor through its <greatest common divisor> with that number.
First double $P$ using the <elliptic-curve addition formula>. Its tangent slope is
$$
m=\frac{3\cdot1^2+5}{2\cdot1}=4,
\qquad x(2P)=4^2-2=14,\qquad
y(2P)=4(1-14)-1=-53.
$$
These calculations are valid modulo $221$ since the only initial denominator is two, which is invertible. Thus $2P=(14,-53)$, equivalently $(14,168)$ modulo $221$.
To form $3P=P+2P$, the chord slope would require
$$
\frac{-53-1}{14-1}=\frac{-54}{13}\pmod{221}.
$$
The denominator has
$$
\boxed{\gcd(13,221)=13,\qquad221=13\cdot17.}
$$
It cannot be inverted, and thirteen is the requested proper factor. The failure has a geometric explanation: modulo thirteen, $2P=(1,-1)=-P$, so $3P$ is the point at infinity. Modulo seventeen that chord denominator remains nonzero. The differing behavior at the two <primes> is exactly what the <elliptic curve method> detects.
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