Mathematics: A Discrete Introduction
3rd Edition
ISBN: 9780840049421
Author: Edward A. Scheinerman
Publisher: Cengage Learning
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Chapter 1.3, Problem 3.12E
a.
To determine
To find: The number of positive divisors.
b.
To determine
To find: The number of positive divisors.
c.
To determine
To find: The number of positive divisors.
d.
To determine
To find: The number of positive divisors.
e.
To determine
To find: The number of positive divisors.
f.
To determine
To find: The number of positive divisors.
g.
To determine
To find: The number of positive divisors.
h.
To determine
To find: The number of positive divisors.
i.
To determine
To find: The number of positive divisors.
j.
To determine
To find: The number of positive divisors.
k.
To determine
To find: The number of positive divisors.
l.
To determine
To find: The number of positive divisors.
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By considering appropriate series expansions,
e². e²²/2. e²³/3.
....
=
= 1 + x + x² + ·
...
when |x| < 1.
By expanding each individual exponential term on the left-hand side
the coefficient of x- 19 has the form
and multiplying out,
1/19!1/19+r/s,
where 19 does not divide s. Deduce that
18! 1 (mod 19).
Proof: LN⎯⎯⎯⎯⎯LN¯ divides quadrilateral KLMN into two triangles. The sum of the angle measures in each triangle is ˚, so the sum of the angle measures for both triangles is ˚. So, m∠K+m∠L+m∠M+m∠N=m∠K+m∠L+m∠M+m∠N=˚. Because ∠K≅∠M∠K≅∠M and ∠N≅∠L, m∠K=m∠M∠N≅∠L, m∠K=m∠M and m∠N=m∠Lm∠N=m∠L by the definition of congruence. By the Substitution Property of Equality, m∠K+m∠L+m∠K+m∠L=m∠K+m∠L+m∠K+m∠L=°,°, so (m∠K)+ m∠K+ (m∠L)= m∠L= ˚. Dividing each side by gives m∠K+m∠L=m∠K+m∠L= °.°. The consecutive angles are supplementary, so KN⎯⎯⎯⎯⎯⎯∥LM⎯⎯⎯⎯⎯⎯KN¯∥LM¯ by the Converse of the Consecutive Interior Angles Theorem. Likewise, (m∠K)+m∠K+ (m∠N)=m∠N= ˚, or m∠K+m∠N=m∠K+m∠N= ˚. So these consecutive angles are supplementary and KL⎯⎯⎯⎯⎯∥NM⎯⎯⎯⎯⎯⎯KL¯∥NM¯ by the Converse of the Consecutive Interior Angles Theorem. Opposite sides are parallel, so quadrilateral KLMN is a parallelogram.
By considering appropriate series expansions,
ex · ex²/2 . ¸²³/³ . . ..
=
= 1 + x + x² +……
when |x| < 1.
By expanding each individual exponential term on the left-hand side
and multiplying out, show that the coefficient of x 19 has the form
1/19!+1/19+r/s,
where 19 does not divide s.
Chapter 1 Solutions
Mathematics: A Discrete Introduction
Ch. 1.1 - Simplify the following algebraic expression:...Ch. 1.2 - Prob. 2.1ECh. 1.3 - Prob. 3.1ECh. 1.3 - Prob. 3.2ECh. 1.3 - Prob. 3.3ECh. 1.3 - Prob. 3.4ECh. 1.3 - Prob. 3.5ECh. 1.3 - Prob. 3.6ECh. 1.3 - Prob. 3.7ECh. 1.3 - Prob. 3.8E
Ch. 1.3 - Prob. 3.9ECh. 1.3 - Prob. 3.10ECh. 1.3 - Prob. 3.11ECh. 1.3 - Prob. 3.12ECh. 1.3 - Prob. 3.13ECh. 1.3 - Prob. 3.14ECh. 1.4 - Prob. 4.1ECh. 1.4 - Prob. 4.2ECh. 1.4 - Prob. 4.3ECh. 1.4 - Prob. 4.4ECh. 1.4 - Prob. 4.5ECh. 1.4 - Prob. 4.6ECh. 1.4 - Prob. 4.7ECh. 1.4 - Prob. 4.8ECh. 1.4 - Prob. 4.9ECh. 1.4 - Prob. 4.10ECh. 1.4 - Prob. 4.11ECh. 1.4 - Prob. 4.12ECh. 1.5 - Prove that the sum of two odd integers is even.Ch. 1.5 - Prove that the sum of an odd integer and an even...Ch. 1.5 - Prove that if n is an odd integer, then n is also...Ch. 1.5 - Prove that the product of two even integers is...Ch. 1.5 - Prove that the product of an even integer and an...Ch. 1.5 - Prove that the product of two odd integers is odd.Ch. 1.5 - Prove that the square of an odd integer is odd.Ch. 1.5 - Prove that the cube of an odd integer is odd.Ch. 1.5 - Suppose a, b, and c are integers. Prove that if ab...Ch. 1.5 - Suppose a, b, and c are integers. Prove that if...Ch. 1.5 - Suppose a, b, d, x, and y are integers. Prove that...Ch. 1.5 - Suppose a, b, c, and d are integers. Prove that if...Ch. 1.5 - Let x be an integer. Prove that x is odd if and...Ch. 1.5 - Let x be an integers. Prove that x is odd if and...Ch. 1.5 - Let x be an integer. Prove that 0x if and only if...Ch. 1.5 - Let a and b be integers. Prove that ab if and only...Ch. 1.5 - Let a be a number with a1. Prove that a number x...Ch. 1.5 - Prove that the difference between consecutive...Ch. 1.5 - Let a be a perfect square. Prove that a is the...Ch. 1.5 - For real numbers a and b, prove that if 0ab, then...Ch. 1.5 - Prove that the difference between distinct,...Ch. 1.5 - Prove that an integer is odd if and only if it is...Ch. 1.5 - Suppose you are asked to prove a statement of the...Ch. 1.5 - Suppose you are asked to prove a statement of the...Ch. 1.6 - Disprove: If a and b are integers with ab, then...Ch. 1.6 - Disprove: If a and b are nonnegative integers with...Ch. 1.6 - Disprove: If a, b, and c are positive integers...Ch. 1.6 - Disprove: If a, b, and c are positive integers,...Ch. 1.6 - Disprove: If p and q are prime, then p+q is...Ch. 1.6 - Disprove: If p is prime, then 2p1 is also prime.Ch. 1.6 - Prob. 6.7ECh. 1.6 - An integer is a palindrome if it reads the same...Ch. 1.6 - Prob. 6.9ECh. 1.6 - Prob. 6.10ECh. 1.6 - Prob. 6.11ECh. 1.6 - Prob. 6.12ECh. 1.6 - Prob. 6.13ECh. 1.7 - Prob. 7.1ECh. 1.7 - Prob. 7.2ECh. 1.7 - Prob. 7.3ECh. 1.7 - Prob. 7.4ECh. 1.7 - Prob. 7.5ECh. 1.7 - Prob. 7.6ECh. 1.7 - Prob. 7.7ECh. 1.7 - Prob. 7.8ECh. 1.7 - Prob. 7.9ECh. 1.7 - Prob. 7.10ECh. 1.7 - Prob. 7.11ECh. 1.7 - Prob. 7.12ECh. 1.7 - Prob. 7.13ECh. 1.7 - Prob. 7.14ECh. 1.7 - Prob. 7.15ECh. 1.7 - Prob. 7.16ECh. 1.7 - Prob. 7.17ECh. 1.7 - Prob. 7.18ECh. 1.7 - Prove that xy can be reexpressed in terms of just ...Ch. 1.7 - Prob. 7.20ECh. 1 - Prob. 1STCh. 1 - Prob. 2STCh. 1 - Prob. 3STCh. 1 - Prob. 4STCh. 1 - Prob. 5STCh. 1 - Prob. 6STCh. 1 - Prob. 7STCh. 1 - Prob. 8STCh. 1 - Prob. 9STCh. 1 - Prob. 10STCh. 1 - Prob. 11STCh. 1 - Prob. 12STCh. 1 - Prob. 13STCh. 1 - Prob. 14STCh. 1 - Prob. 15STCh. 1 - Prob. 16STCh. 1 - Prob. 17STCh. 1 - Prob. 18STCh. 1 - Prob. 19STCh. 1 - Prob. 20ST
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