You are given N distinct numbers. You are tasked with finding the number of groups of 2 or 3 that can be formed whose sum is divisible by three.
Input : 1 5 7 2 9 14 Output : 13 The groups of two that can be formed are: (1, 5) (5, 7) (1, 2) (2, 7) (1, 14) (7, 14) The groups of three are: (1, 5, 9) (5, 7, 9) (1, 2, 9) (2, 7, 9) (2, 5, 14) (1, 9, 14) (7, 9, 14) Input : 3 6 9 12 Output : 10 All groups of 2 and 3 are valid.
Naive Approach : For each number, we can add it up with every other number and see if the sum is divisible by 3. We then store these sums, so that we can add each number again to check for groups of three.
Time Complexity: O(N^2) for groups of 2, O(N^3) for groups of 3
Auxiliary Space: O(N^2)
If we carefully look at every number, we realize that 3 options exist:
- The number is divisible by 3
- The number leaves a remainder of 1, when divided by 3
- The number leaves a remainder of 2, when divided by 3
Now, for groups of two being divisible by 3, either both number have to belong to category 1 (both are divisible by 3), or one number should leave a remainder 1, and the other a remainder 2. This way the remainders add up to 3, making the sum divisible by 3.
To form a group of three, either all three numbers should give the same remainder, or one should give remainder 0, another should give 1, and the last should give 2.
In this way, we do not care about the numbers themselves, but their respective remainders. Thus by grouping them into three categories, we can find the total possible groups using a simple formula.
Let C1 be number of elements divisible by 3.
Let C2 be number of elements leaving remainder 1.
Let C3 be number of elements leaving remainder 2.
Answer = C2 * C3 + C1 * (C1 - 1) / 2 --> Groups of 2 + C1 * (C1 - 1) * (C1 - 2) / 6 + C2 * (C2 - 1) * (C2 - 2) / 6 + C3 * (C3 - 1) * (C3 - 2) / 6 --> Groups of 3 with elements of same remainder + C1 * C2 * C3 --> Groups of three with all distinct remainders
Asked in Amazon
This article is contributed by Aditya Kamath. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.
- Number of ways of distributing N identical objects in R distinct groups with no groups empty
- Count the number of pairs (i, j) such that either arr[i] is divisible by arr[j] or arr[j] is divisible by arr[i]
- Count the number of ways to divide N in k groups incrementally
- Number of ways of distributing N identical objects in R distinct groups
- Maximize number of groups formed with size not smaller than its largest element
- Minimum number of swaps required to make a number divisible by 60
- Number of ways to split a binary number such that every part is divisible by 2
- Check if a large number is divisible by a number which is a power of 2
- Check if a number is divisible by all prime divisors of another number
- Number of digits to be removed to make a number divisible by 3
- Number of substrings with length divisible by the number of 1's in it
- Find if a number is divisible by every number in a list
- Number is divisible by 29 or not
- Number of divisors of a given number N which are divisible by K
- Check if the given number is divisible by 71 or not
- Check if the number is divisible 43 or not
- Check if a number is divisible by 47 or not
- Check if a number is divisible by 31 or not
- Check if a number is divisible by 41 or not
- Check if a number is divisible by 23 or not