Find Minimum Operations to Make All Elements Divisible by Three

Find Minimum Operations to Make All Elements Divisible by Three

What is this problem about?

The Find Minimum Operations to Make All Elements Divisible by Three interview question is a straightforward mathematical simulation. You are given an array of integers. In one operation, you can either add 1 or subtract 1 from any element. Your task is to calculate the total minimum number of operations needed so that every single element in the array is divisible by three.

Why is this asked in interviews?

Companies like Google and Bloomberg use this Find Minimum Operations coding problem as a warm-up or "sanity check" for beginner-level positions. It tests your basic understanding of the modulo operator and your ability to apply a simple rule across an array. It evaluations whether you can write clean, efficient $O(N)$ code for a basic arithmetic task.

Algorithmic pattern used

This problem uses the Math interview pattern and Linear Scan.

1. Iterate: Loop through each number in the array.
2. Modulo: Find the remainder when the number is divided by 3 (num % 3).
3. Decision Logic:
   • If the remainder is 0, the number is already divisible. 0 operations.
   • If the remainder is 1 (e.g., 4, 7, 10), subtracting 1 makes it divisible. 1 operation.
   • If the remainder is 2 (e.g., 5, 8, 11), adding 1 makes it divisible. 1 operation.
4. Summation: Add up these operations for all elements.

Example explanation

Array: [1, 2, 3, 4]

• 1: 1 % 3 = 1. Subtract 1. Operations: 1.
• 2: 2 % 3 = 2. Add 1. Operations: 1.
• 3: 3 % 3 = 0. No action. Operations: 0.
• 4: 4 % 3 = 1. Subtract 1. Operations: 1. Total operations: $1 + 1 + 0 + 1 = 3$.

Common mistakes candidates make

• Adding instead of subtracting: Trying to add 2 to a number with remainder 1. While this works mathematically, it uses 2 operations instead of the 1 operation needed for subtraction.
• Over-complicating: Attempting to use dynamic programming or complex searches for a problem that is entirely independent for each element.
• Looping too much: Trying to actually perform the addition/subtraction in a while loop instead of just calculating the cost using modulo.

Interview preparation tip

Always look for the "cost" formula in math problems. In this case, the cost for any non-divisible number is always 1. Recognition of such patterns shows you can simplify problems before writing code.