Unit Digit Method Number System Unit digit Method Wisdom Helps

📘 Unit Digit Method (Depth & Important Points)

🔹 1. Introduction

The unit digit method is a smart technique used in aptitude and competitive exams to quickly determine the last digit of large numbers. Instead of expanding the entire number, we only focus on the cyclic behavior of the unit digit.

This method is especially useful in problems involving:

• Powers (exponentiation)
• Products of large numbers
• Factorials and expansions
• Remainder-based problems focusing on last digit

🔹 2. Core Concept

• Every digit (0–9) follows a cyclic pattern of unit digits when raised to powers.
• Once the cycle is identified, the required digit is found using modulus of cycle length.

Example:
For 2ⁿ, unit digit cycle = (2, 4, 8, 6), length = 4.
Unit digit of 2⁷⁵ → 75 mod 4 = 3 → same as 2³ = 8.

🔹 3. Important Cyclic Patterns

Base Digit	Cycle of Unit Digits	Cycle Length
0	(0)	1
1	(1)	1
2	(2,4,8,6)	4
3	(3,9,7,1)	4
4	(4,6)	2
5	(5)	1
6	(6)	1
7	(7,9,3,1)	4
8	(8,4,2,6)	4
9	(9,1)	2

🔹 4. Key Applications

1. Finding the last digit of powers
   Example: Unit digit of 7²²² → cycle (7,9,3,1). 222 ÷ 4 → remainder 2 → unit digit = 9.
2. Simplifying products
   Example: (12 × 43 × 57) → (2 × 3 × 7) = 42 → unit digit = 2.
3. Factorials
   From 5! onwards, factorials end in 0 → Unit digit of 100! = 0.
4. Divisibility & remainders
   Helps eliminate options quickly in MCQs.

🔹 5. Advantages

• Saves time in exams.
• Reduces errors by focusing only on unit digit.
• Great for power-based problems.
• Helps in MCQ option elimination.

🔹 6. Professional Tips for Students

• Memorize cycles for digits 2–9.
• Always reduce powers using mod cycle length.
• For factorials ≥ 5!, unit digit = 0.
• Practice with various bases to build speed.
• Be alert if the question asks for last two digits — needs advanced methods.

✅ In Short: The Unit Digit Method is about spotting patterns, cycles, and shortcuts, not lengthy calculations. Once mastered, it becomes one of the fastest tools in an aptitude student’s toolkit.