5.2 notes - 5.2 Divisibility TestsWe say A is divisible by...

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5.2 Divisibility Tests- We say “A is divisible by K” whenever A is a multiple of K, that is, if A= K*a for some whole number a. The following phrases all have the same meaning: 1) A is divisible by K 2) K divides A 3) A is a multiple of K 4) K is a factor of A Divisibility can be illustrated by rectangular arrays. A number is divisible by 4 if A squares can be arranged in a rectangular array with 4 rows of the same length. Example: Is 537 divisible by 7? 537/7=76 R 5. Therefore, it is not divisible by 7 because it has a remainder. The same strategy can be used to separate the given number into a large part, which is clearly divisible by the given number, and a leftover part. (See page 114) LEMMA 2.5 (Divisible Lemma) Suppose A is a number divisible by K. Then B is divisible by K “if and only if” A+B is divisible by K. -Picture proof: pg 114 - Algebraic proof: Because A is divisible by K we can write A=Ka for some whole number a. If B is also divisible by K, then B=Kb for some whole number b. By the distributive property, A+B=ka+kb= k(a+b), so A+B is divisible by k.
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