SOLUTION: 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

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.
RELATED QUESTIONS
(a^m+n)^m-n (answered by Fombitz)

Solve the equation a = m - n for the variable n. A. n = a + m B. n = m - a C. (answered by dkppathak)

Prove that for n<0 (answered by robertb)
(a^m+n)(a^m-n) (answered by algebrahouse.com)
If {{{a^m*a^n}}}={{{a^mn}}},then... (answered by Alan3354)
if n < m and a < b, then a. n+a > m+b b. m=b c. n=a d. n+a <... (answered by tommyt3rd)
a^m*a^n= (answered by nyc_function,stanbon)
Show that if a│m and a│n then... (answered by stanbon)
If m and n are consecutive even integers and m > n > 0, how many integers are greater... (answered by Fombitz)