Question 440609: EXAMPLE:
Theorem: For integers a, m, and n, if a | m and a | n, then a | (m+n).
[The “idea,” just the “bare bones”: m = a·r, n = a·s, m+n = a·r+a·s = a·(r+s). Now add the appropriate words to make it into a readable proof.]
Proof: Given that a | m and a | n, there exist integers r and s such that m = a·r and n = a·s. Adding these two equations gives m+n = a·r+a·s. Factoring out a on the right-hand side gives m+n = a·(r+s). Because r and s are integers, r+s is an integer. Thus from the last equation, m+n equals a times an integer (r+s). It follows from the definition that a | (m+n).
Supply the proof for the following theorem.
Theorem: For integers a and b, if a | b, then a^2 | b^2.
[My “idea”: b = a·r, thus b^2 = (a·r)·( a·r). Now from the associative and commutative laws of multiplication we know that (a·r)·( a·r)= a^2· r^2 (you may so state). Thus b^2 equals a^2 times an integer (why?). The desired result follows from the definition. Now add the appropriate explanations to finish the proof.]
Proof:
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! Hint: Since b^2 = (a*r)(a*r) = a^2*r^2, and r is some integer, what does this say about r^2?
Then, once you finish the proof, I'll let you write it, since it won't be beneficial to you for me to do it. Writing proofs is a skill that takes time to be learned. One tip with writing proofs is, be as concise as possible, write all your steps in sequential order, and make sure your definitions/lemmas can be backed up easily.
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