What is congruence in number theory?
If two numbers and have the property that their difference is integrally divisible by a number (i.e., is an integer), then and are said to be “congruent modulo .” The number is called the modulus, and the statement ” is congruent to (modulo )” is written mathematically as. (1)
What is meant by congruence?
1 : the quality or state of agreeing, coinciding, or being congruent … the happy congruence of nature and reason …— Gertrude Himmelfarb. 2 : a statement that two numbers or geometric figures are congruent.
What is congruence in modular arithmetic?
Modulus congruence means that both numbers, 11 and 16 for example, have the same remainder after the same modular (mod 5 for example). 11 mod 5 has a remainder of 1. 11/5 = 2 R1. 16 mod 5 also has a remainder of 1.
What is congruence in reading?
Definition: Equal in size and shape. Two objects are congruent if they have the same dimensions and shape. Very loosely, you can think of it as meaning ‘equal’, but it has a very precise meaning that you should understand completely, especially for complex shapes such as polygons.
When do you use congruence in number theory?
As with so many concepts we will see, congruence is simple, perhaps familiar to you, yet enormously useful and powerful in the study of number theory. If n is a positive integer, we say the integers a and b are congruent modulo n, and write a ≡ b (mod n), if they have the same remainder on division by n.
Which is the proof of the theorem of congruence?
Theorem 1: Every integer is congruent ( mod m) to exactly one of the numbers in the list :- 0, 1, 2, ……. (m – 2), (m -1). Proof:From a theorem in Divisibility, sometimes called Division Algorithm, for every integer a, there exist unique integers qand rsuch that a = qm + r, with 0 £r
Is the number 6 a congruent number or not?
Similarly, 6 is a congruent number because it is the area of a (3,4,5) triangle. 3 and 4 are not congruent numbers. If q is a congruent number then s2q is also a congruent number for any natural number s (just by multiplying each side of the triangle by s ), and vice versa.
How to know if p is a congruent number?
For example, it is known that for a prime number p, the following holds: 1 if p ≡ 3 ( mod 8), then p is not a congruent number, but 2 p is a congruent number. 2 if p ≡ 5 (mod 8), then p is a congruent number. 3 if p ≡ 7 (mod 8), then p and 2 p are congruent numbers.
