3 Congruence
We read this as “a is congruent to b modulo (or mod) n. The algebra of congruences is sometime referred to as “clock arithmetic.” This example.
3 Congruence
We read this as “a is congruent to b modulo (or mod) n. The algebra of congruences is sometime referred to as “clock arithmetic.” This example.
Modular Arithmetic & Congruences - CIS002-2 Computational
15 nov. 2011 technique called modular arithmetic where we use congruences ... 3 Calculate the least absolute residue of 15 × 59 mod (75).
Junior Mathletes: NSML Meet #3 – Modular Arithmetic
NSML Topic #3: NO CALCULATOR. Modular Arithmetic: May include arithmetic operations in different moduli divisibility
Congruence and Congruence Classes
or adding b + d to both sides of this equation
Math 3527 (Number Theory 1)
Polynomial Congruences Modulo pn and Hensel's Lemma Polynomial Congruences II. Example: Solve the equation x3 + x + 2 ? 0 (mod 36).
Math 255 – Spring 2017 Solving x2 ? a (mod n)
Therefore a/-1 (mod n/) exists and the equation can be solved by division to give a unique solution x/ modulo n/. Then the solutions of the original equation
6 Number Theory II: Modular Arithmetic Cryptography
https://www.math.upenn.edu/~mlazar/math170/notes06.pdf
Number theory
congruences which are mathematical statements used to compare the You'll also find out how modular arithmetic is used to help prevent errors.
Section 2.1: Shift Ciphers and Modular Arithmetic
Example 9: Make a table of y values for the equation y = )5(. + x. MOD 9. Solution: ·. Fact: Solving equations (and congruences) if modular arithmetic is
[PDF] Congruences and Modular Arithmetic - mathillinoisedu
This type of manipulation is called modular arithmetic or congruence magic and it allows one to quickly calculate remainders and last digits of numbers with
[PDF] Modular Arithmetic
Since any two integers are congruent mod 1 we usually require n ? 2 from now on Modular arithmetic is sometimes introduced using clocks
[PDF] 62 Modular Arithmetic - Penn Math
First we can multiply the two numbers directly and obtain 306; some calculation will show that 306 is congruent to 2 modulo 19 Alternatively we know that 17
[PDF] 3 Congruence
The algebra of congruences is sometime referred to as “clock arithmetic ” This example illustrates this Imagine you are a mouse and that each day you
[PDF] Modular Arithmetic
This contradiction shows that the equation has no solutions These examples show that linear congruences may have solutions or may be unsolvable We can under-
[PDF] Congruences
This particular integer is called the modulus and the arithmetic we do with this type of relationships is called the Modular Arithmetic For example the
[PDF] 3 Congruences and Congruence Equations
1 What is the remainder when 17113 is divided by 3? Don't bother asking your calculator: 17113 is 139 digits long! Instead we use modular arithmetic:
[PDF] modular arithmetic - keith conrad
MODULAR ARITHMETIC KEITH CONRAD 1 Introduction We will define the notion of congruent integers (with respect to a modulus) and develop
[PDF] Examples of Modular Arithmetic
We say that a and b are congruent modulo n; we denote a ? b First of all we recall how to solve linear Diophantine equations: Claim 0 (Solving Linear
[PDF] Everything You Need to Know About Modular Arithmetic
Definition Let m > 0 be a positive integer called the modulus We say that two integers a and b are congruent modulo m if b ? a is
Congruences and Modular Arithmetic
Congruences:We sayais congruent tobmodulom, and writeabmodm, ifaandbhave the same remainder when divided bym, or equivalently ifabis divisible bym. Equivalently, the congruence notationabmodmcan be thought of as a shorthand notation for the statement \there exists an integerksuch thata=b+km." Here are some examples to illustrate this notation: (1) 517 mod 3 (since 5 and 17 both have remainder 2 when divided by 3, or equivalently, since 175 = 12 is divisible by 3). (2) 10 4 mod 7 (since 10(4) = 14 is divisible by 7) (3) 20130 mod 3 (since 20130 = 2013 is divisible by 3) (4)n0 mod 2 ifnis even, andn1 mod 2 ifnis odd. (5)ndmod 10 ifnhasdas last decimal digit (since thenn=d+ 10kfor some integerk, and hencendmod 10) Modular arithmetic:The key fact about congruences is thatcongruences to the same modu- lus can be added, multiplied, and taken to a xed positive integral power.For example, since 6 1 mod 7, we have 61000(1)1000= 1 mod 7. This type of manipulation is called modular arithmeticorcongruence magic, and it allows one to quickly calculate remainders and last digits of numbers with thousands of digits. Congruence magic: An example.Consider the problem of nding the last digit of 20132014 (which is a number with thousands of decimal digits). The last decimal digit is the same as the remainder modulo 10, so the problem can be restated as nding 20132014mod 10, i.e., the remainder
of 20132014modulo 10. Here is how to do this with minimal computations using congruence magic:
20133 mod 10 (reduce base mod 10);
2013201432014mod 10;(simplify the problem);
34= 811 mod 10;(nd small power that is 1 mod 10);
34k1k= 1 mod 10 (for any pos. integerk)
32012= 345031 mod 10 (takek= 503 to get an exponent close to 2014)
32014= 3201232132= 9 mod 10
Thus the last digit of 2013
2014is 9.
Fermat's Theorem.An important special congruence is given by Fermat's Theorem, which says that ifpis a prime number andais not divisible byp, thena p11 modp. For example, since 2017 is a prime number, we havea20161 mod 2017 for any integerathat is not a multiple of 2017. Important note: Fermat's theorem only applies if the modulus,p, is a prime number. In particular, the theorem cannot be applied in the above example (which required computing 20132014mod 10) since the modulus, 10, is not a prime number. (In fact, from the above calculation,
we have 39338313 mod 10, whereas Fermat's theorem, if it were applicable, would give
391 mod 10.)
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