8 - LECTURE 8 DIVISIBILITY 4.1-4.2 United we stand...

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Unformatted text preview: LECTURE 8: DIVISIBILITY ( § 4.1-4.2) United we stand, divided we fall. Aesop (620 BC - 560 BC) 1. What does it mean to divide integers? Definition 1. Let a,b ∈ 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 ∈ Z with b = ac . We write a | b . Example 2. Does 4 | 52? Yes, since 52 = 4 · 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 ∈ Z with b = ac . Result. Let a,b,c ∈ Z . If a | b and b | c then a | c . Proof. Assume a | b and b | c . This means there exists integers x,y ∈ Z with b = ax and c = by . Thus c = by = axy = a ( xy ). Since xy ∈ Z we have a | d . Result. Let a,b,c ∈ Z . If a | c and b | d then ab | cd . Proof. Assume a | c and b | d . There there exist integers x,y ∈ Z with c = ax and d = by . Thus cd = axby = ab ( xy ). Since xy ∈ Z we have ab | cd ....
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8 - LECTURE 8 DIVISIBILITY 4.1-4.2 United we stand...

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