Looking for Integers

455x + 1547y = 50000 + n

5 × 91x + 17 × 91y = 91 × 549 + 41 + n

91(5x + 17y) = 91 × 549 + (41 + n)

(41 + n) must be divisible by 91 then 41 + n = 91

then n = 50

First, note that 455 and 1547 are not coprime. GCD of 455 and 1547 is 91. Since x,y,n are integers, then we divide both sides by 91 to get 5x+17y = (50,000 + n)/91 = 549 + (41+n)/91 Since x,y is an integer, 549 + (41+n)/91 also has to be an integer Now, 41+n = 91k, where k is a positive integer So the minimum value of n is 91 x 1 - 41 = 50 Therefore, n is 50 To check, when n is 50, 549 + (41+n)/91 = 550 5x+17y=550 has a solution x=93, y=5 So n is 50.

The equation Ax + By = C has solution if and only if C be dividable by gcd(A, B).

By some calculation we can find that gcd(455, 1547) = 91.

So to find the smallest integer n, we can divide 50,000 by 91 and get the integer part add a one to it and multiple it by 91 so here is the number 50,050. therefore 50,050 is the smallest positive integer so n should be 50.

The gcd of 455 and 1547 is 91. Therefore both sides must be divisible by 91. This gives n>=50. Now for n=50, we have x=93, y=5 is a solution of the equation. Therefore the minimum possible n is 50.

n = 455x + 1547y - 50000 = 91(5x + 17y) - 50000

As 50000 = 91 * 549 + 41, the minimum possible value of n would be when (5x + 17y) is the minimum value >= 550.

5x + 17y = 550 => x = (550 - 17y)/5 = 110 - (17/5)*y This means that 5 should divide y for x to be an integer. Say, y = 5k => x = 110 - 17k.

Minimum possible integral value of n = 91 * 550 - 50000 = 50050 - 50000 = 50.

$455x+1547y=50000+n$ is a Linear Diophantine equation.A linear Diophantine equation is of the form:

$ax+by=c$

where $a$ and $b$ are coefficients of $x$ and $y$ . We are going to use the following lemma to solve the problem at hand.

Lemma:A solution(infinite solutions,to be exact) exists if and only if $gcd(a.b)|c$

Sketch of proof: Let $gcd(a,b)\nmid c$ . But then $gcd(a,b)|ax+by=c$ ,a contradiction.For more on LDE, see here

Now,back to the problem.We want the lowest possible value of $n$ where satisfies

$455x+1547y=50000+n$

By our lemma, $gcd(455,1547)|50000+n\implies 91|50000+n$

To minimize $n$ (which must be positive),we find the multiple of $91$ closest to 50000[note that 50000 itself is not a multiple of 91].As n must be positive,the right hand side of the equation must be greater than 50000.Hence,we want the multiple to be greater than 50000.The closest multiple is $50050$ .Therefore,the following has a solution in intgers.

$455x+1547y=50050\implies 4551x+1547y=50000+50\implies n=50$

Hence the lowest possible value is $50$ .

Given,

455x + 1547y = 50000 + n

since 455 and 1547 have the no. 7 * 13 i.e 91 in common , take out the common factor That becomes,

91(5x + 17y) = 50000 + n or , 5x + 17y = (50000 + n)91

Hence 50000 + n should be divisible by 91 to have (x,y) as integers.

the nearest multiple of 19.. to 50000 is 500050

Hence the value of n = 50

The gcd(455,1547) = 91
So according to Euclidean Algorithm (50000+n) ≡ 0 (mod 91)
We know ,

50000 ≡ 41(mod 91)
but we have (50000+n) ≡ 0 (mod 91)
or, 41 + n ≡ 0 (mod 91)
or, n ≡ - 41 (mod 91)
or, n ≡ 91 - 41 (mod 91) or, n ≡ 50 (mod 91)
as stated the smallest positive integer . so n = 50

Since $gcd(455,1547) = 91$ , -> $50000 + n$ must be divisible by $91$ .

By trial and errors, we get $n_{min} = \boxed{50}$ . ~~~

Note: $50050 = 91\times550$

Let y be 0, because 1547 is bigger than 455, giving a less detailed addition for each multiplication (the number adds up more per each multiplication than for 455). Then we divide 50,000 by 455, the remainder is 405. That means after we subtract 455(x-1) from 50,000, we have 405 left. The equation will be: 455 = 405 + n, so n is $\boxed{50}$

Given equation 455x+1547y = 50,000+n

GCD of 455 and 1547 is found to be 91.

Diving the Equation by 91 gives 5x+13y = 549+((41+n)/91) Since n has to be as small as possible, the possible least of n such that (41+n) is multiple of 91 is n = 50;

So , n= 50 is a solution if 5x+13y = 550 has a solution (x,y) with x and y as integers. Now , the above equation can be written as x+(13/5)y = 110; and so y = (110-x)*(5/13);

If y has to be an integer, (100-x) should be a multiple of 13. One such case is x =6 and y will be 40. There may me many number of cases. So, the solution exists when n =50, the least possible value in this case.

try to multiply both the numbers to reach the given nearest number.

Since positive integer (nonzero) was required I started by multiplying bigger number by 10, 20, 30 to find suitable starting point and then checked my already prepared list of multiples of 455.

No fine art here, just brute force.

The second part of the expression i.e. 1547y has to be as closest as possible to 50 000 . With y = 30 , 1547y = 46410 , so the difference between 50 000 and the expression becomes 3590 . The nearest value of the first part of the expression ie 455x with x as an integer to 3590 is 3640 with x = 8 , Thus for x = 8 and y = 30 , the expression is 50050 resulting in n = 50

let y=0, then we have 455x=50000+n <=> 455 | 50000+n <=> 455 | 109*(455)+405+n so (405+n) must divisible by 455, then we get n=455-405=50.

455x + 1547y = 50,000 + n GCD(455,1547)=91 Then 91 I 50,000 + n.....so we get 50,000 + n = 91 x 549 + (41+n). Must be noticed that (41+n) must be divisible by 91, therefore we get 41+n=91, finally we get n the smallest positive integer is 50

observe that 455+1547=2002 and 5000 would be 2000 25-->(2002-2) 25 --> 2*25 = 50

455 * X + 1547 * Y = GCD (455, 1547) .... (1)

Greatest Common Divisor of 455 and 1547 is 91.

ceil (50000/91) = 550.

Multiplying (1) by 550 we get:

455 * 550 * X + 1547 * 550 * Y = 91 * 550 = 50050 = 50000 + 50

So n = 50.

455/5 = 91

1547/91= 17

50000/91= 549.45

50000+n/91 = 550

n = 50

455x+1547y=50000+n admets a non-empty set of solutions if and only if PGD(455,1547) divides 50 000+n

Which means 91 divides 50 000+n and n is minimal

hence n=50 ( first multiple of 91 that is bigger than 50 000 is 50050)

I don't know if this is a very good solution but it is how I solved the problem. What I did was that I tried to reach as close as possible to 50,000 multiplying 1547 with a possible value of y. Then, I multiplyied 455 with a possible value of x so that when added to 1547*y would exceed 50,000 by the minimum number possible. What I did was this:

I set y=30: 1547*30=46,410.

46,410 is close to 50,000.

I set x=8: 455*8= 3640.

46,410+ 3,640=50.050

So, 455 x + 1547 y = 50,000 + n <=> 50,050 = 50,050 + n <=> 50,050 - 50,000 = n <=> n = 50

455 x 110 = 50050 ---> 50050 - 50000 = 50

Using the Euclidean Algorithm with $(1547, 455)$ , we get

$1547 = 455(3) + 182$

$455 = 182(2) + 91$

$182 = 91(2) + 0$

So, we know that $gcd(1547, 455) = 91$

Now, going backwards, we see that (I've skipped the steps, refer to Extended Euclidean Algorithm )

$91 = 455(7) - 2(1547)$

As $⌈50,000/91⌉ = 550$ , we can multiply $91$ with $550$ to get $50,050$ .

Finally,

$50,050 = 50,000 + n$ , yielding $n = 50$ .

Since the least possible value of $455x + 1547y$ where $(x,y)$ are integers is $91$ and according to Bezout's indentity the solution to this equation is every multiple of $91$ .

And $50,000/91$ is approximately equal to $549$ .

Therefore $50,000 + n = 550×91 = 50,000 + 50$ Hence the answer is $50$ .