(a) Use the division algorithm to show that any number N can be written as either N = 3k, N = 3k + 1, or N = 3k + 2. Use this to show that the product of any three consecutive integers must be divisible by 3. (In fact it must be divisible by 6. Why?) (b) Could there be integers that are neither odd or even? How would you prove that every integer must either be odd or even? [Hint: When we defined an even number we said it was of the form 2k, and an odd number is of the form 2k + 1. It is theoretically possible with this definition that a number is neither odd nor even. i.e. there might be numbers that are not picked up by this definition. Using the division algorithm with divisor 2, show that every integer must be odd or even. As
(a) Use the division algorithm to show that any number N can be written as either N = 3k, N = 3k + 1, or N = 3k + 2. Use this to show that the product of any three consecutive integers must be divisible by 3. (In fact it must be divisible by 6. Why?)
(b) Could there be integers that are neither odd or even? How would you prove that every integer must either be odd or even? [Hint: When we defined an even number we said it was of the form 2k, and an odd number is of the form 2k + 1. It is theoretically possible with this definition that a number is neither odd nor even. i.e. there might be numbers that are not picked up by this definition. Using the division algorithm with divisor 2, show that every integer must be odd or even. As a result of this, it follows that consecutive integers have “opposite parity.” That is, if one is odd, the other is even.]
(a) Use the division algorithm to show that any number N can be written as either N = 3k, N = 3k + 1, or N = 3k + 2. Use this to show that the product of any three consecutive integers must be divisible by 3. (In fact it must be divisible by 6. Why?)
(b) Could there be integers that are neither odd or even? How would you prove that every integer must either be odd or even?
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