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Unformatted text preview: LECTURE 8: DIVISIBILITY ( Â§ 4.14.2) United we stand, divided we fall. Aesop (620 BC  560 BC) 1. What does it mean to divide integers? Definition 1. Let a,b âˆˆ Z . We say a divides b , or a is a divisor of b , or b is a multiple of a if there exists an integer c âˆˆ Z with b = ac . We write a  b . Example 2. Does 4  52? Yes, since 52 = 4 Â· 13. Does 3  13? No, since there is no integer c with 13 = 3 c . In this case we write 3 13. Example 3. If you ever see the statement a  b just turn it into the statement that there exists some c âˆˆ Z with b = ac . Result. Let a,b,c âˆˆ Z . If a  b and b  c then a  c . Proof. Assume a  b and b  c . This means there exists integers x,y âˆˆ Z with b = ax and c = by . Thus c = by = axy = a ( xy ). Since xy âˆˆ Z we have a  d . Result. Let a,b,c âˆˆ Z . If a  c and b  d then ab  cd . Proof. Assume a  c and b  d . There there exist integers x,y âˆˆ Z with c = ax and d = by . Thus cd = axby = ab ( xy ). Since xy âˆˆ Z we have ab  cd ....
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 Spring '11
 Smith
 Division, Remainder, Integers, Euclidean algorithm

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