Exercise 5.2.14. Suppose January 25 is a Thursday. (a) Use Definition 5.2.6 to determine whether January 3 is a Thursday. Show your reasoning. (b) Use Proposition 5.2.10 to determine whether January 31 is a Thursday. Show your reasoning. (c) Find the nearest Thursday to January 15. Show your reasoning.
Exercise 5.2.14. Suppose January 25 is a Thursday. (a) Use Definition 5.2.6 to determine whether January 3 is a Thursday. Show your reasoning. (b) Use Proposition 5.2.10 to determine whether January 31 is a Thursday. Show your reasoning. (c) Find the nearest Thursday to January 15. Show your reasoning.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please do part a and b and please show step by step
Transcribed Image Text:Exercise 5.2.14. Suppose January 25 is a Thursday.
(a) Use Definition 5.2.6 to determine whether January 3 is a Thursday.
Show your reasoning.
(b) Use Proposition 5.2.10 to determine whether January 31 is a Thursday.
Show your reasoning.
(c) Find the nearest Thursday to January 15. Show your reasoning.
(d) Find the nearest Thursday to April 18. Show your reasoning. (Note:
the year is not a leap year.)
Transcribed Image Text:Definition 5.2.6. Two integers a and b are equivalent mod m if both a
and b have the same remainder when divided by m. To denote that a and
b are equivalent mod m, we write: a = b (mod m).
A
Proposition 5.2.10. Given any two integers a and b, and a modulus m (m
is a positive integer). Then
a = b (mod m) if and only if a – b = k - m,
where k is an integer.
We may rewrite Proposition 5.2.10 more elegantly using mathematical short-
hand as follows: Given a, b, m e Z, then
a =b (mod m) iff m|(a – b).
Note the two shorthand expressions we have used here: the symbol 'e'
means 'contained in' or 'elements of", while the single vertical line 'l' means
"divides'.
The following proposition establishes important facts about modular
equivalence that we'll need later.
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