Question

please prove this statement sir
if a and b are prime to one another,then nos;1a,2a,3a....(b-1)a leave different remaining when divided by b

Answer 1

try a proof by contradiction is my first guess.
suppose \(a_i\) and \(a_j\) leave the same remainder when divide by \(b\) , call that remainder \(r\) and see if you can show that that means \(gcd(a, b)\neq 1\)

Answer 2

rather i meant \(ia\) and \(ja\) leave the same remainder
ignore the subscripts, they were stupid

Answer 3

hint, consider \(ia-ja=(i-j)a\)

Answer 4

sir its proof is
if possible let ra and sa leave the same remainders when divided by b,then
ra-sa=m(b)
(r-s)a=m(b)
since a is prime to be
a
not equal to m(b)such tha r-s=m(b) which is contradiction
where
r,s<b
such that r-s<b
so our assemption is wrong
so the remainders are different
pls help me sir for prove this with adding numbers