Question

Select the mistake that is made in the proof given below. Theorem. For any two integers, x and y, if x | y, then 2? | ya. Proof. Let x = 3 and y = 6. Therefore, x and y are both integers. 2? = 32 = 9, and y? = 62 = 36. Since 9 | 36, then x? | y?. Therefore, if zly.then r? | yº Failure to properly introduce a variable. Assuming facts that have not yet been proven. Misuse of existential instantiation Generalizing from examples

Answer 1

The mistake is generalizing the example. We can not say that a theorem is true if it is only true for one example. We have to prove for arbitrary elements. Here it is.

let n and كم we By be two integers. Fundamental theorem of arithentic write product of primes, say, yapabila - Portal can 2 and y as Is dz paip E* where si, ßiyo. So x² = p 20, 2202 229 . Dan and y²= p 28 p2 Since aly, by unique factonization, necessarily disßi t > 2 di 223i , So x² lyz.