In mathematics, the **unit digit** is the digit in the one's place of a number. This simple concept holds the key to a powerful shortcut that can dramatically speed up calculations, especially when dealing with large numbers or complex operations like powers and multiplication. Mastering the unit digit method is a game-changer for competitive exams and mental math.
1. What is the Unit Digit?
The unit digit is the rightmost digit of any integer. For example:
- The unit digit of 567 is 7.
- The unit digit of 1,234 is 4.
- The unit digit of 90 is 0.
The unit digit of a number is determined solely by the unit digits of the numbers being operated on. This is the fundamental principle behind this method.
2. Finding the Unit Digit of a Product
To find the unit digit of a product, you only need to multiply the unit digits of the numbers involved.
Method:
- Identify the unit digit of each number.
- Multiply these unit digits together.
- The unit digit of the result is the unit digit of the final product.
Example 1: Find the unit digit of 43 × 28
Unit digit of 43 is 3.
Unit digit of 28 is 8.
Multiply the unit digits: 3 × 8 = 24.
The unit digit of 24 is 4.
Result: 4
Example 2: Find the unit digit of 123 × 456 × 789
Unit digits are 3, 6, and 9.
Step 1: 3 × 6 = 18. Unit digit is 8.
Step 2: Take the unit digit from the last result (8) and multiply by the next unit digit (9): 8 × 9 = 72.
The unit digit of 72 is 2.
Result: 2
3. Finding the Unit Digit of a Number Raised to a Power
This is where the unit digit method truly shines. Instead of calculating the entire power, we can use a concept called **cyclicity** to find the pattern of the unit digits.
4. The Concept of Cyclicity
The unit digits of powers repeat in a cycle. The length of this cycle is called the **cyclicity**.
Cyclicity of Different Unit Digits:
• Unit Digits 0, 1, 5, 6: Cyclicity of 1
Any number ending in 0, 1, 5, or 6, when raised to any power, will have the same unit digit as the base.
320¹ = 320 → Unit digit is 0
320² = 102400 → Unit digit is 0
...and so on. The same applies to 1, 5, and 6.
• Unit Digits 4, 9: Cyclicity of 2
These digits have a cycle of 2. The unit digit depends on whether the power is even or odd.
**Powers of 4:**
4¹ = 4
4² = 16 (unit digit 6)
4³ = 64 (unit digit 4)
**Pattern: 4, 6, 4, 6...**
**Powers of 9:**
9¹ = 9
9² = 81 (unit digit 1)
9³ = 729 (unit digit 9)
**Pattern: 9, 1, 9, 1...**
**Rule:** If the power is **odd**, the unit digit is the same as the base (4 or 9). If the power is **even**, the unit digit is 6 (for base 4) or 1 (for base 9).
• Unit Digits 2, 3, 7, 8: Cyclicity of 4
These digits have a cycle of 4. We find the remainder of the power when divided by 4.
**Powers of 2:** 2, 4, 8, 6, 2, 4, 8, 6...
**Powers of 3:** 3, 9, 7, 1, 3, 9, 7, 1...
**Powers of 7:** 7, 9, 3, 1, 7, 9, 3, 1...
**Powers of 8:** 8, 4, 2, 6, 8, 4, 2, 6...
**Rule:**
- Divide the power (exponent) by 4 and find the remainder.
- If the remainder is 1, the unit digit is the same as the base (raised to power 1).
- If the remainder is 2, the unit digit is the same as the base² unit digit.
- If the remainder is 3, the unit digit is the same as the base³ unit digit.
- If the remainder is **0** (i.e., the power is a multiple of 4), the unit digit is the same as the base⁴ unit digit.
Example: Find the unit digit of 137⁴⁸
Step 1: The unit digit of the base is 7.
Step 2: The power is 48. Divide 48 by 4.
48 ÷ 4 = 12 with a remainder of 0.
Step 3: Since the remainder is 0, we look at the unit digit of 7⁴.
7¹=7, 7²=49 (unit 9), 7³=343 (unit 3), 7⁴=2401 (unit 1).
The unit digit of 7⁴ is 1.
Result: 1
5. Practice Problems
Test your knowledge with these problems:
1. Unit digit of 25 × 34 × 19 = ?
2. Unit digit of 82²⁴ = ?
3. Unit digit of 124²⁵⁵ = ?
4. Unit digit of 125⁶² + 326²⁴ = ?
5. Unit digit of 999⁹⁹ = ?
6. Conclusion
The unit digit method is a powerful tool in your mental math arsenal. It allows you to quickly find the last digit of a number without performing the full calculation. This is particularly useful for solving problems in competitive exams where answer choices are often distinguished by their unit digits.
Remember these key principles:
- For multiplication, only multiply the unit digits.
- For powers, identify the cyclicity (1, 2, or 4).
- Use the remainder of the power divided by the cyclicity to find the final unit digit.
With practice, these shortcuts will become second nature, helping you save time and boost your confidence in mathematics. Ready to test your skills? Try our interactive games and challenges on MathsGenius!
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