congruences A-Z

this thread is mainly intended to give some insight into congruences and how to solve various jee related problems using that concept..

definition of congruent:

A number `a` is said to be congruent to `b` modulo `m` if `m` divides (a-b)

its written as a ≡ b mod m

so we can say that when we write a≡b mod m ,, `b` is the remainder that is obtained when` a` is divided by `m`.

now let us take a few examples:

9≡1mod2

13≡1mod4

but now u get this doubt, when 13 is divided by 4 , remainder is why only 1 ? why cant we say the remainder is -3 ????

infact this doubt is correct. we can even write

13≡-3mod4

so we can write in various ways as we wish to..

SOME PROPERTIES OF CONGRUENCES:

1) we have say

a≡ bmod p
c≡ k mod p

then we can write,

(a+c)≡ (b+k) mod p

but here u need to observe that the number with which we are dividing a and c is the same number `p`. so that has to be kept in mind..

similarly we can aslo write

(a-c)≡ (b-k) mod p

and ac≡ bk mod p

here also we see that the number with which we are dividing a and c by the same number `p`. so alwyas rememebr that .

now its obvious that division cannot be defined here ( as cogruence itself is a way of decribing the process of division)

so the foloowing properties are trivial to observe:

1) a≡ bmod 0 implies a=b

2) a≡ a mod m

3) a≡ b mod m implies b≡ a mod m

4) a≡ b mod m and b≡ c mod m implies a≡ c mod m

5) a≡ b mod m implies ka≡ kb mod m

6)a≡ b mod m implies aⁿ≡ bⁿ mod m

i think these are enough keeping jee syllabus in mind. now i want to know if u guys have any doubts whatsoever i have posted till now here?

after that we can start with questions