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+#LyX 2.4 created this file. For more info see https://www.lyx.org/
+\lyxformat 620
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\input defs
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children no
+\language english
+\language_package default
+\inputencoding utf8
+\fontencoding auto
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_roman_osf false
+\font_sans_osf false
+\font_typewriter_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\float_placement class
+\float_alignment class
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_formatted_ref 0
+\use_minted 0
+\use_lineno 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style english
+\dynamic_quotes 0
+\papercolumns 1
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+\html_math_output 0
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+\html_be_strict false
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+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+exerc1[10]
+\end_layout
+
+\end_inset
+
+Example (3) shows a situation in which 
+\begin_inset Formula $X_{4}=X_{0}$
+\end_inset
+
+,
+ so the sequence begins again from the beginning.
+ Give an example of a linear congruential sequence with 
+\begin_inset Formula $m=10$
+\end_inset
+
+ for which 
+\begin_inset Formula $X_{0}$
+\end_inset
+
+ never appears again in the sequence.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+answer 
+\end_layout
+
+\end_inset
+
+This happens if 
+\begin_inset Formula $a=c=0$
+\end_inset
+
+ and 
+\begin_inset Formula $0<X_{0}<m$
+\end_inset
+
+;
+ then the sequence starts with 
+\begin_inset Formula $X_{0}\neq0$
+\end_inset
+
+ and then it's constant at 0.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+rexerc2[M20]
+\end_layout
+
+\end_inset
+
+Show that if 
+\begin_inset Formula $a$
+\end_inset
+
+ and 
+\begin_inset Formula $m$
+\end_inset
+
+ are relatively prime,
+ the number 
+\begin_inset Formula $X_{0}$
+\end_inset
+
+ will always appear in the period.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+answer 
+\end_layout
+
+\end_inset
+
+For 
+\begin_inset Formula $a\geq2$
+\end_inset
+
+,
+ by Euler's theorem,
+ 
+\begin_inset Formula $a^{\varphi(m)}\equiv1\mod m$
+\end_inset
+
+,
+ so 
+\begin_inset Formula $X_{\varphi(m)}=X_{0}$
+\end_inset
+
+ by Equation (6).
+ For 
+\begin_inset Formula $a=1$
+\end_inset
+
+,
+ 
+\begin_inset Formula $X_{m}=(X_{0}+mc)\bmod m=X_{0}$
+\end_inset
+
+,
+ and for 
+\begin_inset Formula $a=0$
+\end_inset
+
+,
+ 
+\begin_inset Formula $a$
+\end_inset
+
+ and 
+\begin_inset Formula $m$
+\end_inset
+
+ are not relatively prime.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+exerc3[M10]
+\end_layout
+
+\end_inset
+
+If 
+\begin_inset Formula $a$
+\end_inset
+
+ and 
+\begin_inset Formula $m$
+\end_inset
+
+ are not relatively prime,
+ explain why the sequence will be somewhat handicapped and probably not very random;
+ hence we will generally want the multiplier 
+\begin_inset Formula $a$
+\end_inset
+
+ to be relatively prime to the modulus 
+\begin_inset Formula $m$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+answer 
+\end_layout
+
+\end_inset
+
+Let 
+\begin_inset Formula $p$
+\end_inset
+
+ be a common prime between 
+\begin_inset Formula $a$
+\end_inset
+
+ and 
+\begin_inset Formula $c$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+X_{n}=(a^{n}X_{0}+(a^{n-1}+a^{n-2}+\dots+1)c)\bmod m,
+\]
+
+\end_inset
+
+so 
+\begin_inset Formula 
+\[
+X_{n}\bmod p=c\bmod p
+\]
+
+\end_inset
+
+ for any 
+\begin_inset Formula $n\geq1$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document