Print N lines of 4 numbers such that every pair among 4 numbers has a GCD K
Given N and K, the task is to print N lines where each line contains 4 numbers such that every among those 4 numbers has a GCD K and the maximum number used in N*4 should be minimized.
Note: In case of multiple outputs, print any one.
Input: N = 1, K = 1
Output: 1 2 3 5
Every pair among 1, 2, 3 and 5 gives a GCD K and the largest number among these is 5 which the minimum possible.
Input: 2 2
2 4 6 22
14 18 10 16
In the above input, the maximum number is 22, which is the minimum possible to make 2 lines of 4 numbers.
Approach: The first observation is that if we can solve the given problem for K=1, we can solve the problem with GCD K by simply multiplying the answers with K. We know that any three consecutive odd numbers have a GCD 1 always when paired, so three numbers of every line can be easily obtained. Hence the lines will look like:
1 3 5 _ 7 9 11 _ 13 15 17 _ . . .
An even number cannot be inserted always, because inserting 6 in third line will give GCD(6, 9) as 3. So the best number that can be inserted is a number between the first two off numbers of every line. Hence the pattern looks like:
1 2 3 5 7 8 9 11 13 14 15 17 . . .
To obtain given GCD K, one can easily multiply K to the obtained numbers. Hence for i-th line:
- the first number will be k * (6*i+1)
- the second number will be k * (6*i+1)
- the third number will be k * (6*i+3)
- the fourth number will be k * (6*i+5)
The maximum number among N*4 numbers will be k * (6*i – 1)
Below is the implementation of the above approach.
2 4 6 10 14 16 18 22
Time Complexity: O(N*4), as we are using a loop to traverse N times and doing 4 operations in each traversal.
Auxiliary Space: O(1), as we are not using any extra space.