Equal Remainders

This means that, for some $a,b\in\mathbb{Z}$ , we have: $23-aN=44-bN$ Where, $0\le aN<23$ , $0\le bN<44$ .

Hence: $bN-aN=44-23$ And: $(b-a)N=21$ Clearly, then, N is a maximum, without zero-divisors, when $b-a=1$ . And we are done.

Every number can be uniquely express as the form

$a=bq+r$ where $r$ is the remainder and $r<b$

From the problem, we get:

$23=Nk_1+r$

$44=Nk_2+r$

Given that their remainder is same, we have

$23-Nk_1=44-Nk_2$

$N(k_2-k_1)=21$

For the largest integer $N$ , we let $k_2-k_1$ be 1. Hence, $N=21$

23 = Nk1 + c (1)

44 = Nk2 + c (2)

21 = N(k2-k1)

since k2-k1 is also integer so N is divisible by 21 and

and N < 23 from (1) so N = 21

Let 23≡a(mod k), and 44≡a(mod k). So (44-23)≡(a-a)(mod k) --> 21≡0(mod k) Therefore k = 21.

$mod(23/N)=mod(44/N)$ Therefore $mod((44-23)/N)=0$ $44-23=21$ Largest value of N is 21 that satisfies the above condition

Let the equations be 23 = N * q1 + r and 44 = N * q2 + r.

Where N, q1 and q2 are the quotients and remainder.

Subtracting the above equations we get,

21 = N * (q2 - q1)

As 21 can be factorized as 7 * 3 and 21 * 1 , according to question N should have its maximum value.

Therefore, q2 - q1 = 1 and N = 21 (Ans.)

I started to divide 23 by the largest divisor which is 23 and got no remainder while when 44 is divided by 23, there is a different remainder. I used 22 as a divisor for both and the remainders are not the same. When the divisor is 21, the remainder for both is 2.

Forming 2 equations from the above question: Na+r=23 .....(1) Nb +r=44 .....(2) where, a and b are the numbers with which N is multiplied to get the same remainder Subtracting(2) from (1), we get: N(b-a)=21 Since, we need the largest value of N, so, N=21

i typed this as c++ program

void main(void){ int i; for(i=22;i>1;i--) { cout<<i<<"\t"<<44%i<<"\t"<<23%i<<"\n"; } }

Say

23 = pN + r

and

44 = qN + r

where ( 0<= r < N ) (The division algorithm)

It is clear that q>=p.

Equating r from the two above equations,

N = 21 / (q - p)

Since N has to be positive and we are required to find N max, (q - p) should be the minimum positive integer....(which is 1)

so, N = 21.

Using Python:

for i in range(1,24):
    print(23/i,44/i)

This piece of code divides $23$ and $44$ by each number from 1 to 23,

I then manually checked for the highest value of i where the decimal part of both numbers were equal.