document.write( "Question 440607: Supply the proof for the following theorem.\r
\n" ); document.write( "\n" ); document.write( "Theorem: For integers a, b, and c, if a | b and b | c, then a | c.\r
\n" ); document.write( "\n" ); document.write( "[My “idea”: b = a·r, c = b·s, c = (a·r)·s = a·(r·s). Don't know the rest.]\r
\n" ); document.write( "\n" ); document.write( "Outline: Because a | b and b | c, there exist integers r and s such that
\n" ); document.write( "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.\r
\n" ); document.write( "\n" ); document.write( "Proof: \n" ); document.write( "

Algebra.Com's Answer #304316 by richard1234(7193) $\"\"$ $\"About$

You can put this solution on YOUR website!
From this line:\r
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\n" ); document.write( "\n" ); document.write( "[My “idea”: b = a·r, c = b·s, c = (a·r)·s = a·(r·s). Don't know the rest.] \r
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\n" ); document.write( "\n" ); document.write( "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. \n" ); document.write( "

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