Mastering Unit Digits and Last Two Digits for Competitive Exams

Understanding unit digits and last two digits of numbers, especially large powers, is a fundamental skill for excelling in the quantitative aptitude sections of competitive exams like the CAT. This module will equip you with the concepts and techniques to efficiently solve such problems.

Unit Digits: The Basics

The unit digit of a number is the rightmost digit. When dealing with operations like addition, subtraction, and multiplication, the unit digit of the result depends only on the unit digits of the operands. For example, the unit digit of 123 + 456 is the unit digit of 3 + 6, which is 9. Similarly, the unit digit of 123 * 456 is the unit digit of 3 * 6, which is 8.

What is the unit digit of the product of 789 and 321?

The unit digit of 789 is 9 and the unit digit of 321 is 1. The unit digit of their product is the unit digit of 9 * 1, which is 9.

Unit Digits of Powers: Cyclicity

For powers, the unit digit often follows a cyclical pattern. Let's examine the unit digits of powers of some numbers:

0: 0, 0, 0, ... (Cycle length 1)
1: 1, 1, 1, ... (Cycle length 1)
2: 2, 4, 8, 6, 2, 4, 8, 6, ... (Cycle length 4)
3: 3, 9, 7, 1, 3, 9, 7, 1, ... (Cycle length 4)
4: 4, 6, 4, 6, ... (Cycle length 2)
5: 5, 5, 5, ... (Cycle length 1)
6: 6, 6, 6, ... (Cycle length 1)
7: 7, 9, 3, 1, 7, 9, 3, 1, ... (Cycle length 4)
8: 8, 4, 2, 6, 8, 4, 2, 6, ... (Cycle length 4)
9: 9, 1, 9, 1, ... (Cycle length 2)

To find the unit digit of a number raised to a large power, we only need to consider the unit digit of the base and the exponent modulo the cycle length.

The concept of cyclicity for unit digits can be visualized by plotting the unit digits of powers of a base number. For instance, for the base 3, the unit digits are 3 (for $3^1$ ), 9 (for $3^2$ ), 7 (for $3^3$ ), 1 (for $3^4$ ), and then it repeats. This forms a cycle of length 4: (3, 9, 7, 1). To find the unit digit of $3^{25}$ , we find the remainder of 25 divided by 4, which is 1. Thus, the unit digit is the first element of the cycle, which is 3. This visual representation helps in remembering the patterns.

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Last Two Digits: An Extension

Finding the last two digits of a number is equivalent to finding the remainder when the number is divided by 100. This is a more complex problem than finding the unit digit, often involving the Chinese Remainder Theorem or specific properties of powers modulo 100. However, for certain bases, patterns emerge.

For powers of numbers ending in 0, 1, 5, or 6, the last two digits are predictable. For example, any power of a number ending in 0 (greater than 1) will end in 00. Any power of a number ending in 1 will end in 01. Any power of a number ending in 5 (greater than 1) will end in 25. Any power of a number ending in 6 will end in 76 (for powers greater than 1).

A key observation for last two digits: $76^n$ always ends in 76 for $n \ge 1$ . Also, $25^n$ ends in 25 for $n \ge 2$ . Numbers ending in 00, 01, 25, 76 have special properties when raised to powers.

For other bases, finding the last two digits often involves calculating the number modulo 4 and modulo 25 separately, and then combining the results using the Chinese Remainder Theorem. This is because $100 = 4 imes 25$ , and 4 and 25 are coprime.

Example: Last Two Digits of $3^{20}$

We need to find $3^{20} \pmod{100}$ .

Modulo 4: $3 \equiv -1 \pmod{4}$ . So, $3^{20} \equiv (-1)^{20} \equiv 1 \pmod{4}$ .
Modulo 25: We observe the cycle of powers of 3 modulo 25: $3^1 \equiv 3 \pmod{25}$ $3^2 \equiv 9 \pmod{25}$ $3^3 \equiv 27 \equiv 2 \pmod{25}$ $3^4 \equiv 6 \pmod{25}$ $3^5 \equiv 18 \pmod{25}$ $3^{10} \equiv 18^2 = 324 \equiv 24 \equiv -1 \pmod{25}$ $3^{20} \equiv (-1)^2 \equiv 1 \pmod{25}$ .

Now we have two congruences: $x \equiv 1 \pmod{4}$ $x \equiv 1 \pmod{25}$

Since $x$ leaves a remainder of 1 when divided by both 4 and 25, and 4 and 25 are coprime, $x$ must leave a remainder of 1 when divided by their product, $4 imes 25 = 100$ .

Therefore, $3^{20} \equiv 1 \pmod{100}$ . The last two digits are 01.

What is the unit digit of

7^{100}

The cycle for 7 is (7, 9, 3, 1) with length 4. $100 \div 4 = 25$ with a remainder of 0. When the remainder is 0, we take the last element of the cycle, which is 1.

Key Takeaways and Strategies

Unit Digits: Focus on the unit digit of the base and the exponent modulo the cycle length. Remember the cycles for 0-9.
Last Two Digits: For bases ending in 0, 1, 5, 6, the pattern is simpler. For other bases, consider modulo 4 and modulo 25, and use the Chinese Remainder Theorem if necessary.
Practice: Solve a variety of problems involving unit digits and last two digits to build speed and accuracy.

Learning Resources

Unit Digit of Powers - GeeksforGeeks(blog)

A comprehensive guide explaining the concept of unit digits of powers with examples and shortcuts.

Last Two Digits of Powers - BYJU'S(blog)

Explains how to find the last two digits of powers, including cases involving modulo 100 and the Chinese Remainder Theorem.

Cyclicity of Unit Digits - IndiaBIX(blog)

A clear explanation of the cyclicity of unit digits for numbers 0 through 9 with practice questions.

Quantitative Aptitude - Number Systems (Unit Digit)(blog)

Provides a structured approach to finding unit digits of numbers and their powers.

Finding the Last Two Digits of a Number(video)

A video tutorial demonstrating methods to find the last two digits of numbers, often using modulo arithmetic.

Number Systems for CAT Exam: Unit Digit(video)

A YouTube video specifically tailored for CAT aspirants, focusing on unit digits of powers.

Number Systems - Unit Digit and Last Two Digits(video)

Another helpful video resource covering both unit digits and last two digits with examples.

Number Systems - CAT Quantitative Aptitude(blog)

A comprehensive section on Number Systems for CAT, which includes detailed explanations and practice problems on unit digits.

Chinese Remainder Theorem Explained(wikipedia)

A foundational concept for solving last two digits problems, this resource explains the Chinese Remainder Theorem.

Number Systems - Unit Digit Tricks(blog)

Offers quick tricks and shortcuts for calculating unit digits, useful for time-bound exams.