Master the art of finding the unit digit of massive products, squares, and cubes. Learn the cyclicity rule and solve complex math problems in seconds without a calculator.

In this blog, we will break down how to find the unit digit for:

1. Products of numbers.
2. Squares and Cubes.

The Core Concept: Cyclicity (The 4-Step Dance)

Every digit has a “cycle” of repetition when raised to powers. Instead of calculating the entire power, we look at the remainder when the exponent is divided by the cycle length.

The Master Table (Memorize this!)

Number (Base)	Cycle Length	The Pattern (Unit Digits)
0, 1, 5, 6	1	Always ends with itself (e.g., $6^1=6,6^2=36$61=6,62=36).
4, 9	2	4: 4, 6, 4, 6… 9: 9, 1, 9, 1…
2, 3, 7, 8	4	2: 2,4,8,6 3: 3,9,7,1 7: 7,9,3,1 8: 8,4,2,6

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Part 1: Finding Unit Digits in Products

When multiplying two or more large numbers, the unit digit of the product is simply the unit digit of the product of their unit digits.

Example: Find the unit digit of $345\times 567$.

• Unit digit of 345 is 5.
• Unit digit of 567 is 7.
• Multiply: $5\times 7=35$5×7=35.
• The unit digit is 5.

Advanced (with Powers):
Find the unit digit of ${13}^{456}\times {27}^{891}$.

Step 1: Tackle ${13}^{456}$.

• We care about the unit digit of the base: 3.
• The cycle of 3 is [3, 9, 7, 1] (Length 4).
• Divide the exponent (456) by the cycle length (4).
• $456÷4=114$ remainder 0.
• If the remainder is 0, we take the 4th element in the cycle.
• Unit digit = 1.

Step 2: Tackle ${27}^{891}$.

• Base unit digit: 7.
• Cycle of 7 is [7, 9, 3, 1] (Length 4).
• Divide exponent (891) by 4.
• $891÷4=222$ remainder 3.
• We take the 3rd element in the cycle.
• Unit digit = 3.

Step 3: Multiply the results.

• $1\times 3=3$1×3=3.
• Final Answer: The unit digit is 3.

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Part 2: Unit Digit of Squares

Squares are essentially a number raised to the power of 2. Since the exponent is always fixed (2), we don’t need cyclicity; we just need to know the square of the unit digit.

Rule: Square the unit digit of the base number. The unit digit of that result is the answer.

Examples:

1. What is the unit digit of 127^2?
   • Unit digit of base = 7.
   • $7^2=49$=49.
   • Unit digit is 9.
2. What is the unit digit of 184^2?
   • Unit digit = 4.
   • $4^2=16$42=16.
   • Unit digit is 6.

Quick Cheat Sheet for Squares:

• If the number ends in 1 → Square ends in 1.
• If ends in 2 → Square ends in 4.
• If ends in 3 → Square ends in 9.
• If ends in 4 → Square ends in 6.
• If ends in 5 → Square ends in 25 (Unit digit 5).
• If ends in 6 → Square ends in 6.
• If ends in 7 → Square ends in 9.
• If ends in 8 → Square ends in 4.
• If ends in 9 → Square ends in 1.
• If ends in 0 → Square ends in 0.

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Part 3: Unit Digit of Cubes

Cubes are slightly different. Here, we cube the unit digit of the base number.

Rule: Cube the unit digit of the base number. The unit digit of that result is the answer.

Examples:

• What is the unit digit of ${123}^3$?
• Unit digit of base = 3.
• $3^3=27$=27.
• Unit digit is 7.
• What is the unit digit of 98^3?
  • Unit digit = 8.
  • $8^3=512$.
  • Unit digit is 2.

Quick Cheat Sheet for Cubes:

• If the number ends in 1 → Cube ends in 1.
• If ends in 2 → Cube ends in 8.
• If ends in 3 → Cube ends in 7.
• If ends in 4 → Cube ends in 4.
• If ends in 5 → Cube ends in 5.
• If ends in 6 → Cube ends in 6.
• If ends in 7 → Cube ends in 3.
• If ends in 8 → Cube ends in 2.
• If ends in 9 → Cube ends in 9.
• If ends in 0 → Cube ends in 0.

Fun Fact: Notice that the units digit of a cube is the same as the base number for digits 0, 1, 4, 5, 6, and 9!

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Advanced Tip: The “Power of 4” Rule

For bases ending in 2, 3, 7, and 8, the cycle length is 4.
If the exponent is a multiple of 4, the unit digit is ALWAYS the 4th term in the cycle:

• 2^4 ends in 6
• 3^4 ends in 1
• 7^4 ends in 1
• 8^4 ends in 6

If the exponent gives a remainder of 1, 2, or 3, just pick the corresponding term from the cycle.

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Practice Problems (Test Yourself!)

1. Find the unit digit of ${245}^{105}$.
2. Find the unit digit of ${19}^{21}\times {43}^{42}$.
3. What is the unit digit of ${123456}^2$?
4. What is the unit digit of ${99}^3$?

(Scroll down for answers)

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Answers:

1. 5 (Because any power of a number ending in 5 ends in 5).
2. 4 ( ${19}^{21}$ → cycle of 9 is 9,1 | Exponent 21 is odd → Unit digit 9.
   ${43}^{42}$ → cycle of 3 is 3,9,7,1 | 42 mod 4 = 2 → Unit digit 9.
   $9\times 9=81$9×9=81 → Unit digit 1. Wait, let’s correct that!
   Correction: 19^21 ends in 9. 43^42 ends in 9. 9*9=81. Unit digit is 1.)
3. 6 ( $6^2=36$62=36 ).
4. 9 ( $9^3=729$93=729 ).

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Conclusion

Finding the unit digit isn’t magic; it’s pattern recognition. By understanding the cyclicity of numbers, you can solve complex exponent problems faster than a calculator can display the full number. Practice the table above, and you’ll be a math whiz in no time!