Thus, an object with a momentum of 0 (does or does not have
kinetic energy.
if an object has a momentum of 10 it (does or does not) have
kinetic energy.
how do you know?

At momentum zero, kinetic energy = zero

At momentum 10, kinetic energy can never be equal to zero.

Explanation:

An object is said to have zero momentum when either its mass or its velocity is zero. Since, negetive mass doesn't exist, the velocity should be zero and this case, the object must be at rest. Momentum is zero when change in velocity is zero, but kinetic can never be zero when body is motion or has a momentum of 10.

drying towels at the beach