Unit Digits of Large Exponents

1. The Core Idea: What is Cyclicity?

When you need to find the unit digit of an expression like $347^{75}$, calculating the full value is impossible. The key is to realize that only the unit digit of the base matters. The unit digit of $347^{75}$ will be the same as the unit digit of $7^{75}$.

But even $7^{75}$ is too large. Let's observe the pattern of unit digits for powers of 7:

• $7^1=7$
• $7^2=49$ → unit digit is 9
• $7^3=343$ → unit digit is 3
• $7^4=2401$ → unit digit is 1
• $7^5=16807$ → unit digit is 7 (The pattern repeats!)

The sequence of unit digits is 7, 9, 3, 1. This pattern has a length of 4 and will repeat endlessly. This repeating nature is called cyclicity. For the number 7, the cyclicity is 4.

2. Cyclicity of All Digits (0-9)

To solve these problems quickly, we need to know the cyclicity of each digit from 0 to 9. We can group them into three simple categories.

To explore these patterns, I'd like you to read the following article from GeeksforGeeks. It provides a very clear breakdown.

Here is a summary of those patterns, which you should aim to commit to memory:

Group 1: Cyclicity of 1

• Digits: 0, 1, 5, 6
• Rule: The unit digit is always the digit itself, regardless of the power.
  • Example: The unit digit of $125^{999}$ is 5.

Group 2: Cyclicity of 2

• Digits: 4, 9
• Rule: The unit digit alternates based on whether the power is odd or even.
  • For 4: $4^{\text{odd}}\to 4$, $4^{\text{even}}\to 6$
  • For 9: $9^{\text{odd}}\to 9$, $9^{\text{even}}\to 1$
  • Example: The unit digit of $144^{128}$ (even power) is 6. The unit digit of $49^{53}$ (odd power) is 9.

Group 3: Cyclicity of 4

• Digits: 2, 3, 7, 8
• Rule: The unit digit follows a four-step cycle.
  • 2 → (2, 4, 8, 6)
  • 3 → (3, 9, 7, 1)
  • 7 → (7, 9, 3, 1)
  • 8 → (8, 4, 2, 6)

This table provides a great quick reference.

3. The General Method for Cyclicity 4

For the digits in Group 3 (2, 3, 7, 8), we need a systematic process. Since your background is in Computer Science, you can think of this as a simple algorithm using the modulo operator.

The 4-Step Algorithm:

1. Identify the unit digit of the base number. Let's call it $L$.
2. Identify its cyclicity, which is 4 for this group.
3. Find the remainder when the exponent is divided by 4. Let's call it $R$. So, $R=\text{Exponent}(\mathrm{mod}4)$.
4. The required unit digit is the same as the unit digit of $L^R$.

The Most Important Special Case: Remainder is 0
If the exponent is perfectly divisible by 4 (i.e., the remainder $R$ is 0), you do not use a power of 0. Instead, you use a power equal to the cyclicity itself (which is 4).

This video clearly demonstrates this method and the crucial "remainder 0" rule.

Let's walk through an example: Find the unit digit of $348^{98}$.

1. Unit Digit (L): 8
2. Cyclicity: 4 (from the pattern 8, 4, 2, 6)
3. Remainder (R): We divide the exponent 98 by 4. $98÷4=24$ with a remainder of 2. So, $R=2$.
4. Result: The unit digit is the same as $8^R=8^2=64$. The unit digit is 4.

Now, let's try an example with a remainder of 0: Find the unit digit of $127^{100}$.

1. Unit Digit (L): 7
2. Cyclicity: 4 (from the pattern 7, 9, 3, 1)
3. Remainder (R): We divide the exponent 100 by 4. $100÷4=25$ with a remainder of 0.
4. Result: Since the remainder is 0, we use the cyclicity as the power. The unit digit is the same as $7^4=2401$. The unit digit is 1.

4. Handling Complex Expressions

CAT problems often involve expressions with multiple terms connected by addition, subtraction, or multiplication. The strategy is to find the unit digit of each part separately and then perform the final operation.

Let's watch a video that works through some of these more complex examples.

Example: Find the unit digit of $43^{43}-17^{17}$.

• Part 1: $43^{43}$

  • Unit digit is 3 (cyclicity 4).
  • Remainder of $43÷4$ is 3.
  • Unit digit is that of $3^3=27$, which is 7.

• Part 2: $17^{17}$

  • Unit digit is 7 (cyclicity 4).
  • Remainder of $17÷4$ is 1.
  • Unit digit is that of $7^1=7$, which is 7.

• Combine: We need the unit digit of $7-7$, which is 0.

Conclusion

Congratulations on completing the "Arithmetic Foundations" module! You've built a strong base in number theory, from divisibility and prime factors to remainders and now cyclicity.

Key Takeaways:

• To find the unit digit of $A^B$, you only need the unit digit of $A$.
• Digits can be grouped by their cyclicity:
  • Cyclicity 1 (Always the same): 0, 1, 5, 6
  • Cyclicity 2 (Odd/Even power): 4, 9
  • Cyclicity 4 (Remainder method): 2, 3, 7, 8
• For cyclicity 4, divide the exponent by 4. The unit digit is $(\text{last digit}{)}^{\text{remainder}}$.
• Crucial Rule: If the remainder is 0, use the cyclicity (4) as the power, not 0.

In our next lesson, we will begin a new module: Commercial Mathematics. We'll start with the fundamental skill of performing rapid conversions between percentages, fractions, and ratios, a cornerstone for solving problems on profit & loss, interest, and more.