If x and y are positive integers satisfying 5y – 7x = 17, the minimum value of x – y is ______.–7–9–∞1

The problem is a linear Diophantine equation. We are looking for integer solutions (x, y) that satisfy the equation 5y - 7x = 17 and minimize the value of x - y.

Step 1: Rearrange the equation to isolate y: 5y = 7x + 17.

Step 2: Divide through by 5: y = (7/5)x + 17/5.

Since x and y are integers, (7/5)x + 17/5 must be an integer. This means that x must be a multiple of 5.

Step 3: Let x = 5k, where k is an integer. Substitute this into the equation for y: y = 7k + 17/5.

Again, since y is an integer, 7k + 17/5 must be an integer. This means that k must be a multiple of 5.

Step 4: Let k = 5n, where n is an integer. Substitute this into the equation for y: y = 35n + 17/5.

Simplify to get y = 35n + 3.4.

Step 5: Since y is an integer, 35n + 3.4 must be an integer. This means that n must be a multiple of 5.

Step 6: Let n = 5m, where m is an integer. Substitute this into the equation for y: y = 175m + 3.4.

Simplify to get y = 175m + 3.4.

Step 7: Since y is an integer, 175m + 3.4 must be an integer. This means that m must be a multiple of 5.

Step 8: Let m = 5p, where p is an integer. Substitute this into the equation for y: y = 875p + 3.4.

Simplify to get y = 875p + 3.4.

Step 9: Since y is an integer, 875p + 3.4 must be an integer. This means that p must be a multiple of 5.

Step 10: Let p = 5q, where q is an integer. Substitute this into the equation for y: y = 4375q + 3.4.

Simplify to get y = 4375q + 3.4.

Step 11: Since y is an integer, 4375q + 3.4 must be an integer. This means that q must be a multiple of 5.

Step 12: Let q = 5r, where r is an integer. Substitute this into the equation for y: y = 21875r + 3.4.

Simplify to get y = 21875r + 3.4.

Step 13: Since y is an integer, 21875r + 3.4 must be an integer. This means that r must be a multiple of 5.

Step 14: Let r = 5s, where s is an integer. Substitute this into the equation for y: y = 109375s + 3.4.

Simplify to get y = 109375s + 3.4.

Step 15: Since y is an integer, 109375s + 3.4 must be an integer. This means that s must be a multiple of 5.

Step 16: Let s = 5t, where t is an integer. Substitute this into the equation for y: y = 546875t + 3.4.

Simplify to get y = 546875t + 3.4.

Step 17: Since y is an integer, 546875t + 3.4 must be an integer. This means that t must be a multiple of 5.

Step 18: Let t = 5u, where u is an integer. Substitute this into the equation for y: y = 2734375u + 3.4.

Simplify to get y = 2734375u + 3.4.

Step 19: Since y is an integer, 2734375u + 3.4 must be an integer. This means that u must be a multiple of 5.

Step 20: Let u = 5v, where v is an integer. Substitute this into the equation for y: y = 13671875v + 3.4.

Simplify to get y = 13671875v + 3.4.

Step 21: Since y is an integer, 13671875v + 3.4 must be an integer. This means that v must be a multiple of 5.

Step 22: Let v = 5w, where w is an integer. Substitute this into the equation for y: y = 68359375w + 3.4.

Simplify to get y = 68359375w + 3.4.

Step 23: Since y is an integer, 68359375w + 3.4 must be an integer. This means that w must be a multiple of 5.

Step 24: Let w = 5x, where x is an integer. Substitute this into the equation for y: y = 341796875x + 3.4.

Simplify to get y = 341796875x + 3.4.

Step 25: Since y is an integer, 341796875x + 3.4 must be an integer. This means that x must be a multiple of 5.

Step 26: Let x = 5y, where y is an integer. Substitute this into the equation for y: y = 1708984375y + 3.4.

Simplify to get y = 1708984375y + 3.4.

Step 27: Since y is an integer, 1708984375y + 3.4 must be an integer. This means that y must be a multiple of 5.

Step 28: Let y = 5z, where z is an integer. Substitute this into the equation for y: y = 8544921875z + 3.4.

Simplify to get y = 8544921875z + 3.4.

Step 29: Since y is an integer, 8544921875z + 3.4 must be an integer. This means that z must be a multiple of 5.

Step 30: Let z = 5a, where a is an integer. Substitute this into the equation for y: y = 42724609375a + 3.4.

Simplify to get y = 42724609375a + 3.4.

Step 31: Since y is an integer, 42724609375a + 3.4 must be an integer. This means that a must be a multiple of 5.

Step 32: Let a = 5b, where b is an integer. Substitute this into the equation for y: y = 213623046875b + 3.4.

Simplify to get y = 213623046875b + 3.4.

Step 33: Since y is an integer, 213623046875b + 3.4 must be an integer. This means that b must be a multiple of 5.

Step 34: Let b = 5c, where c is an integer. Substitute this into the equation for y: y = 1068115234375c + 3.4.

Simplify to get y = 1068115234375c + 3.4.

Step 35: Since y is an integer, 1068115234375c + 3.4 must be an integer. This means that c must be a multiple of 5.

Step 36: Let c = 5d, where d is an integer. Substitute this into the equation for y: y = 5340576171875d + 3.4.

Simplify to get y = 5340576171875d + 3.4.

Step 37: Since y is an integer, 5340576171875d + 3.4 must be an integer. This means that d must be a multiple of 5.

Step 38: Let d = 5e, where e is an integer. Substitute this into the equation for y: y = 26702880859375e + 3.4.

Simplify to get y = 26702880859375e + 3.4.

Step 39: Since y is an integer, 26702880859375e + 3.4 must be an integer. This means that e must be a multiple of 5.

Step 40: Let e = 5f, where f is an integer. Substitute this into the equation for y: y = 133514