Proof by exhaustion List all the possible cases

Proof by exhaustion • List all the possible cases • Show the statement is true for every case WB 7 Suppose x and y are positive odd integers less than 7. Prove that their sum is divisible by 2. Possible odd integers are 1, 3, 5, Cases are: 1 +3 = 4 which is even 1 + 5 = 6 which is even 3 + 5 = 8 which is even So the sum of two positive odd integers below 7 is divisible by two QED! ‘I’ve proved it !’

WB 8 a Prove that the sum of the squares of any three consecutive odd numbers is always 11 more than a multiple of 12 Three consecutive odd numbers are and The squares of consecutive odd numbers are and

WB 8 b Prove that the sum of the squares of any three consecutive odd numbers is always 11 more than a multiple of 12 So the sum of the squares of three consecutive odd numbers is ‘always 11 more than a multiple of 12’…? ‘I’ve proved it !’

WB 9 the sum is now a b c c b a a b-1 c+10 c b a

WB 9 b the sum is now a-1 b +9 c+10 c b a The multiples of 99 (below 1000) are 99, 198, 297, 396, 495, 693, 792, 891, 990 These always have a digit sum of 18 e. g. 297 2+9+7 = 18 ‘If a three digit number has a digit sum of 18 it is divisible by 99’ The digit sum of our answer above is So the difference between N and K is a multiple of 99 Similarily b) is TRUE ‘I’ve proved it !’

WB 10 Consider any finite list of prime numbers p 1, p 2, . . . , pn P is not prime since it has n divisors (divisible by every integer up to pn ) If Q is prime, then there is at least one more prime than is in the list. This means that at least one more prime number exists beyond those in the list. QED! ‘I’ve proved it !’

WB 11

Disproof by counter example WB 12 a) Identify the error in the students working b) Provide a counter example to show the statement is not true

KUS objectives BAT construct proofs self-assess One thing learned is – One thing to improve is –

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