First day of college: "prove that 0 times x is equal to 0". That got rid of everyone who wasn't really into maths :-)
That exercise was literally the first question, on the first hour of back to school, after 5 minutes of a welcome speech.
I can't even figure out the proof anymore, although it really can't be complicated given the ridiculously small amount of tools you are given (basically x + -x is 0, and (a+b) times c equals a times c + b times c. You are given nothing else.
by commutativity of multiplication, distributivity, the existence of a neutral element 0 for addition, and the commutativity of multiplication once again. Thus, we have that
x*0 + x*0 = x*0.
For clarity, define
A = x*0.
Then we have that
A + A = A
A + A + (-A) = A + (-A), by the existence of additive inverses.
A + (A + (-A)) = (A + (-A)), by associativity of addition
A + 0 = 0, by the definition of additive inverses
A = 0, by the definiton of the neutral element 0 of addition
That exercise was literally the first question, on the first hour of back to school, after 5 minutes of a welcome speech.
I can't even figure out the proof anymore, although it really can't be complicated given the ridiculously small amount of tools you are given (basically x + -x is 0, and (a+b) times c equals a times c + b times c. You are given nothing else.