Given two numbers ‘a’ and ‘b’. Find the minimum difference between any terms in shifted infinite tables of ‘a’ and ‘b’, given shifts ‘x’ and ‘y’, where x, y >= 0.

Let us consider a = 6 and b = 16. Infinite tables of these numbers are.

Table of a : 6, 12, 18, 24, 30, 36, 42, **48.**….

Table of b : 16, 32, **48**, 64, 80, 96, 112, 128…..

Let given shifts be x = 5 and y = 2

Shifted Table of a : 11, **17,** 23, 29, **35**, 41, 47 …

Shifted Table of b : **18**, **34**, 50, 66, 82, 98, 114 …

The minimum difference between any two terms of above two shifted tables is 1 (See colored terms). So the output should be 1.

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**Algorithm:**

- Compute the Greatest Common Divisor (GCD) of the numbers ‘a’ and ‘b’. Let the GCD of two numbers be ‘g’.
- Compute the absolute difference “diff” of the shifts ‘x’ and ‘y’ under modulo ‘g’, i.e., diff = abs(a – b) % g
- The required shifted difference is minimum of ‘diff’ and ‘g – diff’.

Below is implementation of above algorithm.

## C++

`// C++ program to find the minimum difference ` `// between any two terms of two tables ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Utility function to find GCD of a and b ` `int` `gcd(` `int` `a, ` `int` `b) ` `{ ` ` ` `while` `(b != 0) ` ` ` `{ ` ` ` `int` `t = b; ` ` ` `b = a % b; ` ` ` `a = t; ` ` ` `} ` ` ` `return` `a; ` `} ` ` ` `// Returns minimum difference between ` `// any two terms of shifted tables of 'a' ` `// and 'b'. 'x' is shift in table of 'a' ` `// and 'y' is shift in table of 'b'. ` `int` `findMinDiff(` `int` `a, ` `int` `b, ` `int` `x, ` `int` `y) ` `{ ` ` ` `// Calculate gcd of a nd b ` ` ` `int` `g = gcd(a,b); ` ` ` ` ` `// Calculate difference between x and y ` ` ` `int` `diff = ` `abs` `(x-y) % g; ` ` ` ` ` `return` `min(diff, g - diff); ` `} ` ` ` `// Driver Code ` `int` `main() ` `{ ` ` ` `int` `a = 20, b = 52, x = 5, y = 7; ` ` ` ` ` `cout << findMinDiff(a, b, x, y) << endl; ` ` ` ` ` `return` `0; ` `} ` |

## Java

`// Java program to find the ` `// minimum difference between ` `// any two terms of two tables ` `import` `java.io.*; ` ` ` `class` `GFG ` `{ ` ` ` `// Utility function to ` `// find GCD of a and b ` `static` `int` `gcd(` `int` `a, ` `int` `b) ` `{ ` ` ` `while` `(b != ` `0` `) ` ` ` `{ ` ` ` `int` `t = b; ` ` ` `b = a % b; ` ` ` `a = t; ` ` ` `} ` ` ` `return` `a; ` `} ` ` ` `// Returns minimum difference ` `// between any two terms of ` `// shifted tables of 'a' and ` `// 'b'. 'x' is shift in table ` `// of 'a' and 'y' is shift in ` `// table of 'b'. ` `static` `int` `findMinDiff(` `int` `a, ` `int` `b, ` ` ` `int` `x, ` `int` `y) ` `{ ` ` ` `// Calculate gcd ` ` ` `// of a nd b ` ` ` `int` `g = gcd(a,b); ` ` ` ` ` `// Calculate difference ` ` ` `// between x and y ` ` ` `int` `diff = Math.abs(x - y) % g; ` ` ` ` ` `return` `Math.min(diff, g - diff); ` `} ` ` ` `// Driver Code ` `public` `static` `void` `main (String[] args) ` `{ ` ` ` `int` `a = ` `20` `, b = ` `52` `, x = ` `5` `, y = ` `7` `; ` ` ` `System.out.println( findMinDiff(a, b, x, y)); ` `} ` `} ` ` ` `// This code is contributed by ajit ` |

## Python3

`# python3 program to find the minimum difference ` `# between any two terms of two tables ` `import` `math as mt ` ` ` `# Utility function to find GCD of a and b ` `def` `gcd(a,b): ` ` ` ` ` `while` `(b !` `=` `0` `): ` ` ` `t ` `=` `b ` ` ` `b ` `=` `a ` `%` `b ` ` ` `a ` `=` `t ` ` ` ` ` `return` `a ` ` ` `# Returns minimum difference between ` `# any two terms of shifted tables of 'a' ` `# and 'b'. 'x' is shift in table of 'a' ` `# and 'y' is shift in table of 'b'. ` `def` `findMinDiff (a, b, x, y): ` ` ` `# Calculate gcd of a nd b ` ` ` `g ` `=` `gcd(a,b) ` ` ` ` ` `# Calculate difference between x and y ` ` ` `diff ` `=` `abs` `(x` `-` `y) ` `%` `g ` ` ` ` ` `return` `min` `(diff, g ` `-` `diff) ` ` ` `# Driver Code ` `a,b,x,y ` `=` `20` `,` `52` `,` `5` `,` `7` ` ` `print` `(findMinDiff(a, b, x, y)) ` `#This code is contributed by Mohit kumar 29 ` |

## C#

`// C# program to find the minimum difference ` `// between any two terms of two tables ` `using` `System; ` `class` `GFG ` `{ ` ` ` `// Utility function to find GCD of a and b ` `static` `int` `gcd(` `int` `a, ` `int` `b) ` `{ ` ` ` `while` `(b != 0) ` ` ` `{ ` ` ` `int` `t = b; ` ` ` `b = a % b; ` ` ` `a = t; ` ` ` `} ` ` ` `return` `a; ` `} ` ` ` `// Returns minimum difference between any ` `// two terms of shifted tables of 'a' and ` `// 'b'. 'x' is shift in table of 'a' and ` `// 'y' is shift in table of 'b'. ` `static` `int` `findMinDiff(` `int` `a, ` `int` `b, ` ` ` `int` `x, ` `int` `y) ` `{ ` ` ` `// Calculate gcd ` ` ` `// of a nd b ` ` ` `int` `g = gcd(a, b); ` ` ` ` ` `// Calculate difference ` ` ` `// between x and y ` ` ` `int` `diff = Math.Abs(x - y) % g; ` ` ` ` ` `return` `Math.Min(diff, g - diff); ` `} ` ` ` `// Driver Code ` `static` `void` `Main() ` `{ ` ` ` `int` `a = 20, b = 52, x = 5, y = 7; ` ` ` `Console.WriteLine(findMinDiff(a, b, x, y)); ` `} ` `} ` ` ` `// This code is contributed by mits ` |

## PHP

`<?php ` `// PHP program to find the minimum ` `// difference between any two terms ` `// of two tables ` ` ` `// Utility function to ` `// find GCD of a and b ` `function` `gcd(` `$a` `, ` `$b` `) ` `{ ` ` ` `while` `(` `$b` `!= 0) ` ` ` `{ ` ` ` `$t` `= ` `$b` `; ` ` ` `$b` `= ` `$a` `% ` `$b` `; ` ` ` `$a` `= ` `$t` `; ` ` ` `} ` ` ` `return` `$a` `; ` `} ` ` ` `// Returns minimum difference ` `// between any two terms of ` `// shifted tables of 'a' and ` `// 'b'. 'x' is shift in table ` `// of 'a' and 'y' is shift in ` `// table of 'b'. ` `function` `findMinDiff(` `$a` `, ` `$b` `, ` `$x` `, ` `$y` `) ` `{ ` ` ` `// Calculate gcd of a nd b ` ` ` `$g` `= gcd(` `$a` `, ` `$b` `); ` ` ` ` ` `// Calculate difference ` ` ` `// between x and y ` ` ` `$diff` `= ` `abs` `(` `$x` `- ` `$y` `) % ` `$g` `; ` ` ` ` ` `return` `min(` `$diff` `, ` `$g` `- ` `$diff` `); ` `} ` ` ` `// Driver Code ` `$a` `= 20; ` `$b` `= 52; ` `$x` `= 5; ` `$y` `= 7; ` ` ` `echo` `findMinDiff(` `$a` `, ` `$b` `, ` `$x` `, ` `$y` `), ` `"\n"` `; ` ` ` `// This code is contributed by aj_36 ` `?> ` |

**Output :**

2

**Complexity:** O(log n), to compute the GCD.

**How does this work?**

After shifting tables become (Assuming y > x and diff = y-x)

a+x, 2a+x, 3a+x, .. ab+x, …. and b+y, 2b+y, 3b+y, …., ab+y, ….

In general, difference is, abs[(am + x) – (bn + y)] where m >= 1 and n >= 1

An upper bound on minimum difference is “abs(x – y)”. We can always get this difference by putting m = b and n = a.

How can we reduce the absolute difference below “abs(x – y)”?

Let us rewrite “abs[(am + x) – (bn + y)]” as abs[(x – y) – (bn – am)]

Let the GCD of ‘a’ and ‘b’ be ‘g’. Let us consider the table of ‘g’. The table has all terms like a, 2a, 3a, … b, 2b, 3b, … and also the terms (bn – am) and (am – bn).

This article is contributed by **Vaibhav Agarwal**. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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