WJEC Mathematics for AS Level - Pure

15 Test yourself Hence we have 3 cases to test which will cover all the possible integers so we use proof by exhaustion Case 1: If n = 3 p , then n 3 = 27 p 3 , which is a multiple of 9. Case 2: If n = 3 p − 1, then n 3 = 27 p 3 − 27 p 2 + 9 p − 1, which is 1 less than a multiple of 9. Case 3: If n = 3 p − 2, then n 3 = 27 p 3 − 54 p 2 + 36 p − 8 = 27 p 3 − 54 p 2 + 36 p − 9 + 1, which is 1 more than a multiple of 9. So the statement has been proved true for all integers. Test yourself ➊ Prove that if n is an integer where, 2 ≤ n ≤ 7 then n 2 + 2 is not divisible by 4. ➋ Show, by counter-example, that the statement ‘If | a + 1| = | b + 1| , then a = b ’ is false. ➌ Using proof by deduction, prove that for any four consecutive integers, the difference in the products of the last two numbers and the product of the ϐirst two numbers is equal to the sum of all four integers. ➍ Using disproof by counter-example prove that the proposition ‘ if a , b are real numbers, then a + b ≥ 2 √ ab ’ is false. ➎ Using proof by deduction prove that the statement ‘every perfect square is either a multiple of 3 or one more than a multiple of 3’ is true. ➏ Disprove each of the following statements by counter-example. (a) For all real numbers a and b , if b 2 > a 2 , then b > a . (b) For all real numbers x , y and z , if x > y , then xz > yz . ➐ Disprove the following statement by giving a counter-example. If n is an integer and n 2 is divisible by 4, then n is divisible by 4. The two vertical lines show that we take the modulus of the value. The modulus of a value will always change a negative value into a positive value. For example |−3 + 1| = |−2| = 2.