Find the smallest positive integer n such that the equation 455x+1547y=50,000+n has a solution (x,y) where both x and y are integers

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@hans how did you answer this?

To solve the equation 455x + 1547y = 50,000 + n, where both x and y are integers, we need to find the smallest positive integer value of n such that the equation has a solution.

This equation can be rewritten as follows: 1547y = 50,000 + n - 455x.

Let's start by examining the coefficient of x, which is -455. To determine if there is a solution, we need to find the greatest common divisor (GCD) of -455 and 1547.

Using the Euclidean algorithm, we can calculate the GCD as follows:

GCD(1547, -455) = GCD(-455, 1547 mod -455) = GCD(-455, 182) = GCD(-455 mod 182, 182) = GCD(-91, 182 mod -91) = GCD(-91, 0) = 91.

Since the GCD is 91, we know that -455 and 1547 are divisible by 91. Therefore, n must also be divisible by 91 for a solution to exist.

To find the smallest positive integer value of n, we need to find the smallest positive multiple of 91 that is greater than or equal to 50,000. Dividing 50,000 by 91 gives us approximately 549.45.

Therefore, the smallest positive integer n that satisfies the equation is 91 * (integer part of 549.45 + 1) = 91 * 550 = 50,050.

So, the smallest positive integer n is 50,050.