Mathematical Proof: Proving by Implication and Congruent Modulo

Mathematical Proof (Question 1)
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Table of Contents
INTRODUCTION............................................................................................................................. 3
MATHEMATICAL PROOFS.............................................................................................................. 4
Definition 1.................................................................................................................................4
Definition 2.................................................................................................................................4
Definition 3.................................................................................................................................4
Definition 4.................................................................................................................................4
Proving by Implication................................................................................................................4
CONGRUENT (modulo m)...............................................................................................................5
Definition 5.................................................................................................................................5
Examples.................................................................................................................................... 5
Definition 6.................................................................................................................................5
THE PROPOSITION......................................................................................................................... 6
Proof.......................................................................................................................................... 6
Conclusion..................................................................................................................................... 7
References..................................................................................................................................... 8

INTRODUCTION
In this report, we look at explanations to mathematical concepts. As we interact with
mathematical concepts, we are bound to make mathematical statements. For example, the
statement “7 is a prime number” is a mathematical statement. These mathematical statements
are called propositions. A proposition can either be true or false. If a proposition is considered
true then it is called an axiom. Mathematical proofs play a key role in this study.
This paper begins by introducing mathematical proofs and as well gives some definitions that
are necessary for the buildup.
Secondly, the concept of congruent modulo is introduced. Two integers numbers are said to be
congruent modulo a particular natural number if they give the same remainder when divided
by the natural number.
Thirdly, we discuss the body of the paper. This paper discusses a proposition whose prove is
given using the proof by implication method. The proposition says that: If a ≡ c mod mand
b ≡ d mod m then a+ b ≡c +d mod m and ab ≡ cd mod m. That is, if a−c is divisible by m and b−d
is divisible by m, we show that ( a+ b )−(c +d ) is also divisible by m as well as ab−cd .
Finally, a conclusion is given. The conclusion involves explaining the results in the proof in
layman’s language.