| # 571_E. Geometric Progressions | |
| ## Problem Description | |
| Geometric progression with the first element a and common ratio b is a sequence of numbers a, ab, ab2, ab3, .... | |
| You are given n integer geometric progressions. Your task is to find the smallest integer x, that is the element of all the given progressions, or else state that such integer does not exist. | |
| Input | |
| The first line contains integer (1 ≤ n ≤ 100) — the number of geometric progressions. | |
| Next n lines contain pairs of integers a, b (1 ≤ a, b ≤ 109), that are the first element and the common ratio of the corresponding geometric progression. | |
| Output | |
| If the intersection of all progressions is empty, then print - 1, otherwise print the remainder of the minimal positive integer number belonging to all progressions modulo 1000000007 (109 + 7). | |
| Examples | |
| Input | |
| 2 | |
| 2 2 | |
| 4 1 | |
| Output | |
| 4 | |
| Input | |
| 2 | |
| 2 2 | |
| 3 3 | |
| Output | |
| -1 | |
| Note | |
| In the second sample test one of the progressions contains only powers of two, the other one contains only powers of three. | |
| ## Contest Information | |
| - **Contest ID**: 571 | |
| - **Problem Index**: E | |
| - **Points**: 2750.0 | |
| - **Rating**: 3200 | |
| - **Tags**: math | |
| - **Time Limit**: {'seconds': 1, 'nanos': 0} seconds | |
| - **Memory Limit**: 256000000 bytes | |
| ## Task | |
| Solve this competitive programming problem. Provide a complete solution that handles all the given constraints and edge cases. |