I have written the question is obviously to calculate Rc, but in class, after obtaining Ic, we uased the statement Vce = Vc, and just calculated Rc with ohm's law.
Vce is not Vc because the emitter current causes the emitter to have a voltage that is in series with Vce.
It is a stupid assignment because nobody makes a transistor with a beta of 100. Beta is a range of numbers probably from 50 to 200.
Also, nobody biases a transistor with a single resistor to the supply like that.
Assume that the emitter current is 3.65mA. Then the base voltage is about 4.3V and the base current is 36.5uA. The collector current is 3.687mA. You want the collector voltage to be 10V so Rc also has 10V across it and its value is 10V/3.687ma= 2712 ohms.