Proof by exhaustion

A student was asked to prove that the square of any number can be expressed in the form 5n or 5n ± 1 where n The start of the student’s proof is shown in the box below.

Identify and correct an algebraic error in the box above.

Show the calculations and statements that are required to complete the proof.

Use proof by exhaustion to show that for n ₃ , n_n≤ 4

Given that m³ + 5 is odd, use proof by contradiction to show, using algebra, that m is even.

Use proof by exhaustion to show that for n ₃ , n_n≤ 4

Given that m³ + 5 is odd, use proof by contradiction to show, using algebra, that m is even.

Use algebra to prove that the square of any natural number is either a multiple of 3 or one more than a multiple of 3

Prove that for all n , n² + 2 is not divisible by 4

“Given x , the value of 3x – 28 is greater than or equal to the value of (x – 9).” State, giving a reason, if the above statement is always true, sometimes true or never true.

“If m and n are irrational numbers, where m n, then mn is also irrational.”

Disprove this statement by means of a counter example.

Sketch the graph of y x 3

Explain why x 3 x 3 for all real values of x.