5. (FP1.73) Let a and b be positive integers such that a2 and a is even. Then 8

Question

5. (FP1.73) Let a and b be positive integers such that a2 and a is even. Then 8 divides a. We will prove this in steps. (a) Come up with an example of positive integers a and b such that a and a is even. (b) Now prove the statement. You may use the work from class, but your proof should be written out as paragraphs and displayed math that flows. It should not be numbered substeps. In other words, you need to take the work from class and build a coherent proof from it. As a reminder, the steps from class were: i. Given a, b E Z if a2 = b3 and a is even, then 4 divides a. ii. Given that we now know that 4 divides a and that a2-, prove that 4 divides b. ii. Given that 4 divides b and a-, prove that 8 divides a.

Explanation / Answer

(a) example: take a= 8 and b= 4 , here 82 = 43 , also a= 8 is even.

(b) First of all we will prove that for a,b positive integers, if a2 = b3 and a is even then 4 divides a. Since a is even so let a = 2m, for some positive integer m. Now a2 = b3 gives b = cube root of (4m2) , now since our b is a positive integer so there must be 2 as a factor in m2 so in m(otherwise the cube root of 4m2 will not be a positive integer). Hence let m = 2n for some positive integer n. This gives a = 4n,thus 4 divides a.

Now we have a2 = b3 and 4 divides a. Let us now show that 4 divides b. We have b = cube root of a2 . And a = 4x for some positive integer x. So b= cube root of 4×4×k×k. Now for getting the cube root as a positive integer(since b is positive integer) there must be 4 as a factor in k2 . So the value of b must be of the form 4t for some positive integer t. Hence 4 divides b.

In final step, we have a2 = b3 and 4 divides b, now to show 8 divides a. Since a = (b×b×b) and b= 4y for sosome positive integer y. Hence a = (4y×4y×4y) and since a is a integer so the right side must be perfect square hence a = 8z for some positive integer z. Hence 8 divides a. Q.E.D

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