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   Problem Statement  

 Problem Statement for GCDLCM2

Problem Statement

    

Some positive integers, not necessarily distinct, are written on a blackboard. You are given these integers in a format that is specified at the end of this statement.

You are allowed to change the numbers on the blackboard in a sequence of steps. In each step, you have to execute the following actions, in order:

  1. Choose two numbers x and y on the blackboard.
  2. Erase x and y. (Erase exactly two numbers. If there are other copies of these numbers on the blackboard, leave them untouched.)
  3. Write two new integers onto the blackboard: gcd(x,y) and lcm(x,y).
(Above, gcd(x,y) denotes the greatest common divisor and lcm(x,y) the least common multiple of x and y.)

You may perform arbitrarily many steps (possibly even zero). Your goal is to maximize the sum of numbers written on the blackboard. Let S be the largest possible sum. Compute and return the value (S modulo 1,000,000,007).

You are given the int[]s start, d, and cnt, each with the same number of elements. Use the following pseudocode to generate the numbers on the blackboard:

L = length(start)
for i = 0 .. L-1:
    for j = 0 .. cnt[i]-1:
        write the number (start[i] + j * d[i]) onto the blackboard
 

Definition

    
Class:GCDLCM2
Method:getMaximalSum
Parameters:int[], int[], int[]
Returns:int
Method signature:int getMaximalSum(int[] start, int[] d, int[] cnt)
(be sure your method is public)
    
 

Notes

-Note that you are maximizing the sum S. You are not maximizing the return value.
 

Constraints

-start, d and cnt will have the same number of elements.
-start will contain between 1 and 500 elements, inclusive.
-Each element of start and cnt will be between 1 and 10,000,000, inclusive.
-Each element of d will be between 0 and 10,000,000, inclusive.
-For each valid i, start[i] + d[i] * (cnt[i] - 1) will be at most 10,000,000.
-The sum of all cnt[i] will be between 1 and 100,000, inclusive.
 

Examples

0)
    
{1}
{1}
{3}
Returns: 8

There are three numbers on the blackboard: 1, 2 and 3. You can replace numbers 2 and 3 with numbers 1 and 6.

Then sum is 1 + 1 + 6 = 8 (which can be proved to be maximal). The answer is (8 modulo 1,000,000,007) = 8.

1)
    
{3}
{0}
{5}
Returns: 15
There are five numbers 3 on the blackboard. It's impossible to change anything by performing described operations so the maximal sum is 15.
2)
    
{2}
{2}
{4}
Returns: 32
Numbers on the blackboard are 2, 4, 6, 8.
3)
    
{1, 2}
{2, 3}
{2, 2}
Returns: 33
Numbers on the blackboard are 1, 3, 2, 5.
4)
    
{5 ,6}
{23, 45}
{50000, 50000}
Returns: 804225394

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This problem was used for:
       Single Round Match 683 Round 1 - Division I, Level Two