It abounds in problems that yet simple to state, are very hard to solve. This characteristic changes drastically, however, as soon as division is introduced. Prove that the cube of any integer has one of the forms: $7k,$ $7k+1,$ $7k-1.$, Exercise. Let $a$ and $b$ be integers. Division by a negative integer yields a negative remainder. If $a$ and $b$ are integers with $a\neq 0,$ we say that $a$ divides $b,$ written $a | b,$ if there exists an integer $c$ such that $b=a c.$, Here are some examples of divisibility$3|6$ since $6=2(3)$ and $2\in \mathbb{Z}$$6|24$ since $24=4(6)$ and $4\in \mathbb{Z}$$8|0$ since $0=0(8)$ and $0\in \mathbb{Z}$$-5|-55$ since $-55=11(-5)$ and $11\in \mathbb{Z}$$-9|909$ since $909=-101(-9)$ and $-101\in \mathbb{Z}$. Therefore, $k+1\in P$ and so $P=\mathbb{N}$ by mathematical induction. Number Theory. These are notes on elementary number theory; that is, the part of number theory which does not involves methods from abstract algebra or complex variables. First we prove existence. Exercise. It also follows that if it is possible to divide two numbers $m$ and $n$ individually, then it is also possible to divide their sum. An algorithm describes a procedure for solving a problem. Lemma. The theorem does not tell us how to find the quotient and the remainder. If $c|a$ and $c|b,$ then $c|(x a+y b)$ for any positive integers $x$ and $y.$. The total number of times b was subtracted from a is the quotient, and the number r is the remainder. The study of the integers is to a great extent the study of divisibility. Copyright © 2021 Dave4Math LLC. It is not actually an algorithm, but this is this theorem’s If $a|b,$ then $a^n|b^n$ for any natural number $n.$. A division algorithm is an algorithm which, given two integers N and D, computes their quotient and/or remainder, the result of Euclidean division. [June 28, 2019] These notes were revised in Spring, 2019. Thus, if we only wish to consider integers, we simply can not take any two integers and divide them. The notion of divisibility is motivated and defined. The notes contain a useful introduction to important topics that need to be ad-dressed in a course in number theory. The theorem is frequently referred to as the division algorithm (although it is a theorem and not an algorithm), because its proof as given below lends itself to a simple division algorithm for computing q and r (see the section Proof for more). His work helps others learn about subjects that can help them in their personal and professional lives. Prove that $7^n-1$ is divisible by $6$ for $n\geq 1.$, Exercise. Prove or disprove with a counterexample. Its handiness draws from the fact that it not only makes the process of division easier, but also in its use in finding the proof of the Fundamental Theory of Arithmetic. In number theory, we study about integers, rational and irrational, prime numbers etc and some number system related concepts like Fermat theorem, Wilson’s theorem, Euclid’s algorithm etc. Zero is divisible by any number except itself. Definition. David Smith is the CEO and founder of Dave4Math. Division algorithms fall into two main categories: slow division and fast division. Not to be confused with Euclid's division lemma, Euclid's theorem, or Euclidean algorithm. The proof of the Division Algorithm illustrates the technique of proving existence and uniqueness and relies upon the Well-Ordering Axiom. A number other than1is said to be aprimeif its only divisors are1and itself. [thm4] If a, b, c, m and n are integers, and if c ∣ a and c ∣ b, then c ∣ (ma + nb). Show that if $a$ is an integer, then $3$ divides $a^3-a.$, Exercise. Browse other questions tagged elementary-number-theory proof-explanation or ask your own question. 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