Discover product of GCDs of all doable row-column pair of given Matrix

Given a matrix of dimension N*M, the duty is to search out the product of all doable pairs of (i, j) the place i and j are the row quantity and column quantity respectively.

Be aware: Because the reply might be very giant output the reply modulo 1000000007.

Examples:

Enter: N = 5, M = 6
Output: 5760
Clarification: The values of GCD of every doable pair

1	1	1	1	1	1
1	2	1	2	1	2
1	1	3	1	1	3
1	2	1	4	1	2
1	1	1	1	5	1

The product of grid = 1*1*1*1*1*1*1*2*1*2*1*2*1*1*3*1*1*3*1*2*1*4*1*2*1*1*1*1*5*1 = 5760

Enter: N = 34, M = 46
Output: 397325354

Naive Method: To unravel the issue traverse all of the doable pairs of row and column and discover the GCD of them and multiply them with the required reply.

Comply with the steps talked about under to implement the thought:

• Initialize a variable ans = 1 to retailer the product.
• Iterate from i = 1 to N:
  • For every worth of i traverse from 1 to M.
  • Calculate the GCD of every pair.
  • Multiply this with ans.
• Return the ultimate worth of ans because the required reply.

Under is the implementation of the above strategy.

Time Complexity: O(N*M*log(min(N, M)))
Auxiliary House: O(1)

Environment friendly Method: To unravel the issue comply with the under concept:

It may be noticed that for each row, a sample is fashioned until the row quantity and after that, the identical sample repeats.

1	1	1	1	1	1
1	2	1	2	1	2
1	1	3	1	1	3
1	2	1	4	1	2
1	1	1	1	5	1

For instance within the above grid of 4 rows and 6 columns

In row 1, all of the values are 1
In row 2, until index 2 a sample is fashioned and after that very same sample repeats
In row 3, until index 3 a sample is fashioned and after that very same sample repeats

Related observations might be made for all different rows.

Therefore for each row, we solely want to search out the sample as soon as and multiply that sample energy the variety of occasions it happens. This may be completed utilizing Modular exponentiation technique. And eventually we have to multiply the remaining sample energy that is the same as Mpercenti for i^th row.

Additionally, we will take into account the row because the minimal of N and M to scale back time complexity additional.

Under is the implementation of the above strategy:

Time Complexity: min(N, M)*min(N, M)*log(min(N, M))
Auxiliary House: O(1)