Let a and b be integers, and let m be a positive integer. Then a ≡ b (mod m) if and only
if a mod m = b mod m. Prove that:

Question

Let a and b be integers, and let m be a positive integer. Then a ≡ b (mod m) if and only
if a mod m = b mod m. Prove that:

hartnn · Answer

let $a mod m = x$, $b mod m = y$
also x =y , y-x= 0 

so , a = pm +x
similarly 
b = qm +y

subtract these 2 equations 
b-a = (p-q)m + y-x

let p-q =r

b= mr +a

giving 
$a\equiv b~ mod~ m $

hartnn · Answer

@ganeshie8 @mathmath333 
check whether i did it right :P

hartnn · Answer

not sure whether its complete 
*if and only if*

ganeshie8 · Answer

that looks good to me for backward direction

ganeshie8 · Answer

$\impliedby : $
$$a \pmod{m} = b\pmod {m} \implies   a\equiv b \pmod{m}$$

$\implies : $
$$a\equiv b \pmod{m} \implies a \pmod{m} = b\pmod {m} $$

mathmath333 · Answer

b-a = (p-q)m + y-x

this should be

b-a = (q-p)m + y-x

hartnn · Answer

oh right!

hartnn · Answer

forward dir. : 

we have a $\equiv $ b mod m 
b = mr +a 

what next ?

mathmath333 · Answer

^
that is the definition of 

$b\equiv a(mod~ m)$

hartnn · Answer

O.o

when b is divided by m, the remainder is a 
so, 
a $\equiv $ b mod m

hartnn · Answer

or did i get that wrong :P

mathmath333 · Answer

well these are the theorms and its  standard proof u will find in number theory books

mathmath333 · Answer

and that proofs are long i think

hartnn · Answer

i just want to know 
a ≡ b mod m

gives 
b = mr + a
or not ...

mathmath333 · Answer

$14\equiv 4 (mod 10)\~\
4=-10\times 1+14$

anonymous · Answer

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