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.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Transcribed Image Text: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
=
= 3
by factoring out a common factor v
Because x is an integer, so is
and thus n equals 3 · (an integer).
Hence, by definition of divisibility, n is divisible by 3.
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