Find the mistakes in the “proofs” 42. Theorem: The sum of any two even integers

ID: 3119985 • Letter: F

Question

Find the mistakes in the “proofs”

42. Theorem: The sum of any two even integers equals 4k for

some integer k.

“Proof: Suppose m and n are any two even integers. By

definition of even, m = 2k for some integer k and n = 2k

for some integer k. By substitution,

m + n = 2k + 2k = 4k.

This is what was to be shown.”

28. Suppose a, b, c, and d are integers and a = c. Suppose also

that x is a real number that satisfies the equation

ax + b

cx + d = 1.

Must x be rational? If so, express x as a ratio of two integers.

Assumptions made in the given solution where

m = 2k & n = 2k

=> m = n

Which means addition of an even number to itself, which will obviously divisible by 4.

What we need to consider is as,

m = 2k1 & n = 2k2

=> m+n = 2k1 +2k2 = 2(k1+k2)

m+n will be divisible by 4, if k1 & k2 are both odd or both even.

m+n won't be divisible by 4, if k1 & k2 are even & odd or odd or even respectively.

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