Introductory Combinatorics
Introductory Combinatorics
5th Edition
ISBN: 9780134689616
Author: Brualdi, Richard A.
Publisher: Pearson,
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Chapter 6, Problem 1E
To determine

The number of integers between 1 and 10,000 inclusive that are not divisible by 4,5, or 6.

Expert Solution & Answer
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Answer to Problem 1E

The number of integers between 1 and 10,000 inclusive that are not divisible by 4,5, or 6 is 5,334_.

Explanation of Solution

Suppose the set S={1,2,...,104}.

Let A, B, C be the set of integers S that are divisible by 4, 5, 6 respectively.

Use inclusion/exclusion to calculate (A¯B¯C¯) which is determine the number of integers between 1 and 10,000 inclusive that are not divisible by 4,5, or 6.

Where B,C denotes the set of integers in S which are divisible by 5,6 respectively.

Then, |A¯B¯C¯|=|S|(|A|+|B|+|C|)+(|AB| +|BC|+|CA|)(|ABC|).

Where |AB| is the number which is factor by L.C.M of 4,5=20, |BC| is the number which is factor by L.C.M of 5,6=30, |CA| is the number which is factor by L.C.M of 6,4=12 and |ABC| is the number which is factor by L.C.M of 4,5,6=60.

Which is define in given below table;

set size justification
S 104 104×1
A 2500 104=2500×4
B 2000 104=2500×5
C 1666 104=1666×6+4
AB 500 104=500×20
CB 333 104=333×30+10
CA 833 104=833×12+4
ABC 166 104=166×60+40

Table 1

From the table all value substitute in the first formula:

|A¯B¯C¯|=|S|(|A|+|B|+|C|)+(|AB| +|BC|+|CA|)(|ABC|)=104250020001666+500+333+833166=5,334

Since The number of integers between 1 and 10,000 inclusive that are not divisible by 4,5, or 6 is 5,334_.

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Chapter 6 Solutions

Introductory Combinatorics

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