Consider a string consisting of a's, b's, and c's, where the number of b's is three times the number of a's and the number of c's is five times the number of a's. Prove that the length of the string is divisible by 3 Proof: Suppose s is a string of length n that consists of a's, b's, and c's, where the number of b's is times the number of a's, and the number of c's is | times the number of a's. Let x, y and z be the numbers of a's, b's, and c's in s, respectively. The length of s is the sum of the numbers of a's, b's and c's that are in s. Hence, n = x +y + = x+ 3x + by substitution by combining like terms by factoring out a common factor v Because x is an integer, so is x, and thus n equals 3· (an integer). Hence, by definition of divisibility, n is divisible by 3.

Consider a string consisting of a's, b's, and c's, where the number of b's is three times the number of a's and the number of c's is five times the number of a's. Prove that the length of the string is divisible by 3 Proof: Suppose s is a string of length n that consists of a's, b's, and c's, where the number of b's is times the number of a's, and the number of c's is | times the number of a's. Let x, y and z be the numbers of a's, b's, and c's in s, respectively. The length of s is the sum of the numbers of a's, b's and c's that are in s. Hence, n = x +y + = x+ 3x + by substitution by combining like terms by factoring out a common factor v Because x is an integer, so is x, and thus n equals 3· (an integer). Hence, by definition of divisibility, n is divisible by 3.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.2: Arithmetic Sequences

Problem 52E

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