# Modular operation

Is (a/b)mod m == (a mod (b*m)) / b (found it in some solutions of FLOORI4 problem)
Elaborate how’s this possible (Not via example)

Hello Nidhi ,

So,Let us consider …

((a/b) mod m) = k where k<m note this we are gonna use this soon (equation 1)

It means a/b is must be of the form z*m + k for some z>=0

so we have a/b = z*m + k

multiplying both sides by b. we have

a = zmb + k*b

agree till this ?

now if we take mod with b*m on both sides …

a mod bm = (zmb + kb) mod bm

as mod is distributive over the addition operation … we can write RHS of above equation as

a mod bm = (zmb) b*m + (k*b) bm

zmb is a multiple of bm so it would be zero upto modulo with bm

we have
a mod bm = (kb) % b*m

for equation 1 we know that k<m
so bk < mb where b !=0

there we can say that

a mod bm = kb
or (a mod b*m)/b = k

This is what we want to have … if you find anything incorrect or missing and do comment…

2 Likes

Totally Understood

Thanx for appreciation

nice explanation.tnx bro