Moreover, there are many other applications of number theory within the cs fields. In this week we will discuss integer numbers and standard operations on them. A positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. The number is divisible if and only if the result is divisible. As an easy example, note that the sum of the digits of 5823 is equal to. Congruence arithmetic can be used to proof certain divisibility tests. Casting out nines a test for divisibility is called casting out nines.
Using modular arithmetic to prove divisibility math help forum. Applications of modular arithmetic are given to divisibility tests and to block ciphers in cryptography. If you have any integer number it is divisible by 5, if and only if, its last digit is 0 or 5. A congruence can by multiplied through including the modulus. For example, in classic arithmetic, adding a positive number a to another number b always produces a number larger than b. In particular, well get applications to divisibility tests, necessary conditions for solutions of various diophantine equations including nonsolvability. Modular arithmetics pdf the expression a b mod n, pronounced a is congruent to b modulo n, means that mike holt pdf a b is a multiple of n. Use the divisibility tests above to nd which of 2, 4, 5, 7, and 11 divide the following numbers. And the definition of a proof here is basically the same. Finally, we show that, under given addition and multiplication operations, zmz is a ring for any positive integer m. For integers \a\ne 0\ and \b\, we will say that \a\ divides \b\ and write \a\mid b\ if there is an integer \c\ such that \bac\. We say that a divides b write a jb i there exists c 2z such that b ac. Jan 01, 2015 we start number theory by introducing the concept of divisibility and do some simple proofs.
Find the integers c with 0 c 12 such that a c 9a mod. Using these operations is said to be doing arithmetic modulo m. The number should have 0, 2, 4, 6, 0, \ 2, \ 4, \ 6, 0, 2, 4, 6, or 8 8 8 as the units digit. Divisibility tests modular arithmetic may be used to show the validity of a number of common divisibility tests.
Now we know that, since, we have mod and so we have for every. Olympiad number theory through challenging problems. The systematic study of number theory was initiated around 300b. Using modular arithmetic, show that 3 divides n if and only if 3 divides the sum of the digits of n. We say \ a divides b if there is c 2z such that b ac.
Modular arithmetic is sometimes introduced using clocks. Thanks for contributing an answer to mathematics stack exchange. Only the remainder mod 12 is used for time in hours. Again, we can just look at the definition of divisibility. If we depart at 5 oclock and our journey takes 8 hours, we arrive at 1 oclock. Truncate the last digit, multiply it by 5 and subtract from the remaining leading number. The latter operation is the most interesting one and creates a complicated structure on integer numbers. Cs 70 discrete mathematics and probability theory fall 2009 satish rao, david tse note 5 modular arithmetic one way to think of modular arithmetic is that it limits numbers to a prede. Alternate proof using modular arithmetic math notebook. The division algorithm modular arithmetic applications of. Cs 70 discrete mathematics and probability theory modular.
Divisibility in this chapter, we will explore divisibility, the building block of number theory. We know that a a 0, and one of the elementary results seen previously is that n j0 for any. For integers \a e 0\ and \b\, we will say that \a\ divides \b\ and write \a\mid b\ if there is an integer \c\ such that \bac\. Since m clearly divides mi j, we get that m divides r 1 r 2. Using division with remainder in z, there are q and r in z such that. Sep 11, 2016 modular arithmetic and divisibility number theory annie xu and emily zhu september 11, 2016 1 solutions note.
We can also use this proof for the divisibility rule for 9 since 10. The best way to introduce modular arithmetic is to think of the face of a clock. Let a and b be integers, and let m be a positive integer. Modular arithmetic victor adamchik fall of 2005 plan 1.
The proofs of these three parts, while elementary, will provide us with a good exercise in proofwriting. Proof using modular arithmetic this section will illustrate the basic method. Note that any integer is congruent to either 0 or 1 modulo 2. Modulararithmetic modular arithmetic is a way of systematically ignoring di. Number theory, in turn, is foundational for cryptography, the science of keeping ones communications and data secure from eavesdropping third parties. The sum of digits of the number must be divisible by 3 3 3. We write a b to say that a divides b, and a b to say that a does not divide b.
The following requires a basic grounding in modular arithmetic. And in the general case, we have the following, if b divides a, and we have some integer c, then b a c. Applications of modular arithmetic are given to divisibility tests and to block. Truncate the last digit, multiply it by 2 and add to the remaining leading number. So, 3 a again, satisfies the definition of divisibility by b. For example, you may have learned that a natural number is divisible by 9 if the sum of its digits is divisible by 9. The number formed by the tens and units digit of the number must be divisible by. Modular arithmetic and elementary algebra 1 euclids algorithm. But avoid asking for help, clarification, or responding to other answers. Arithmetic mod can be used to give an easy proof of this criterion. An integer a is said to be divisible by some nonzero integer b if there exists an integer c such that a bc. The modular arithmetic lemma handles the cases of addition, subtraction, and multiplication mod n. Congruences are an important and useful tool for the study of divisibility.
Using modular arithmetic to prove divisibility math help. Modular arithmetic in this chapter, well look at some applications of modular arithmetic, i. Many principles of number theory can be leveraged within computation based implementations resulting in not only correct but efficient computations. Modular arithmetics pdf modular arithmetics pdf modular arithmetics pdf download.
Introduction to modular arithmetic 2 number theory basics. We will make ample use of all of these properties from now on. Modular arithmetic motivates many questions that dont arise when studying classic arithmetic. Be sure to keep the properties of divisibility above as well as the definition of congruence modulo n in mind. Modular arithmetic lets us carry out algebraic calculations on. A less known one is the rule for divisibility by 11, which says that a number is divisible by 11 when the number obtained by starting with the ones digit, subtracting the 10s digit, adding the 100s digit, and so on, alternating between adding and subtracting, until youve. First, there is the fact that congruence modulo n satis es 3 popular properties of relations. Use modular arithmetic to prove that this test works. In some sense, modular arithmetic is easier than integer artihmetic because there are only finitely many elements, so to find a solution to a problem you can always try every possbility. If nis an integer, two integers are equal mod nif they di. It is the study of numbers and various ways in which we can combine. Thus, modular arithmetic gives you another way of dealing with divisibility relations. Divisibility and modular arithmetic are foundational to number theory, which is the theory of the integers. The notion of divisibility is one of the most basic properties of the integers definition.
We start number theory by introducing the concept of divisibility and do some simple proofs. In fact, this is the example we shall need for the rsa cryptosystem. Divisibility and modular arithmetic why study number theory. This chapter will introduce many important concepts that will be used throughout the rest of the book. There are several useful properties of modular arithmetic. Modular arithmetic in this week we will discuss integer numbers and standard operations on them. Suppose that the baseten representation of is, where is a digit for each. As to where to begin to prove it yourself, do you know modular arithmetic. Mar 22, 2015 this feature is not available right now. Modular arithmetic and divisibility number theory annie xu and emily zhu september 11, 2016 1 solutions note. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Primes and composites every positive integer n except 1 has at least two positive divisors. Modular arithmetic has many of the same prop erties as.