SOLUTION: Supply the proof for the following theorem. Theorem: For integers a, b, and c, if a | b and b | c, then a | c. [My �idea�: b = a�r, c = b�s, c = (a�r)�s = a�(r�s). Don't

Question 440607: Supply the proof for the following theorem.
Theorem: For integers a, b, and c, if a | b and b | c, then a | c.
[My �idea�: b = a�r, c = b�s, c = (a�r)�s = a�(r�s). Don't know the rest.]
Outline: Because a | b and b | c, there exist integers r and s such that
b = a�r and c = b�s. Then you must explain how these two equations can be combined to give c = (a�r)�s (hint: �substitute a�r for b ��). Use the associative property of multiplication to obtain c = a�(r�s). Now c equals a times an integer (why?). Finish the proof as above.
Proof:
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
From this line:

[My �idea�: b = a�r, c = b�s, c = (a�r)�s = a�(r�s). Don't know the rest.]

You've essentially proven the statement. a|c if and only if c = a*k, where k is some integer. Since c = a*(r*s), and r*s is an integer, then c is equal to a times some integer, a|c, QED.
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