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

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

Word: Because the reply may 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 concept:

• 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.

Beneath is the implementation of the above strategy.

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

Environment friendly Method: To unravel the issue observe the under thought:

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

Comparable observations may be made for all different rows.

Therefore for each row, we solely want to seek out the sample as soon as and multiply that sample energy the variety of instances it happens. This may be performed utilizing Modular exponentiation methodology. And at last 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 cut back time complexity additional.

Beneath is the implementation of the above strategy:

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