SSC MTS Maths Unit Digit in Number System

The Unit Digit is one of the most important topics in the Number System section of SSC MTS Maths. It helps students quickly find the last digit of large calculations involving addition, subtraction, multiplication, powers, and factorials without performing complete calculations.

In competitive exams, questions on unit digits are frequently asked because they test calculation speed and logical thinking. By understanding cyclicity rules, power patterns, and factorial-based shortcuts, candidates can solve unit digit questions accurately in just a few seconds.

Also Read: SSC MTS Syllabus

What is a Unit Digit?

The Unit Digit refers to the last digit of any number. For instance, in 12345, the unit digit is 5. Here are some key characteristics of the unit digit:

• Possible Values: A unit digit can be any single digit from 0 to 9, totaling 10 possible unit digits.

• Non-Negative: A unit digit cannot be a negative number.

• Single Digit: A unit digit must always be a single digit.

Dependency of Unit Digit

The unit digit of a result in arithmetic operations (addition, subtraction, multiplication) always depends solely on the unit digits of the numbers involved.

Unit Digit in Multiplication

Example 1: Find the unit digit of 123 × 26 × 231.

Unit digits: 3, 6, 1.

Product: 3 × 6 × 1 = 18.

The unit digit is 8.

Unit Digit in Addition

Example 2: Find the unit digit of 123 + 12567.

Unit digits: 3, 7.

Sum: 3 + 7 = 10.

The unit digit is 0.

Unit Digit in Subtraction: Special Case

If the unit digit of the first number is smaller than the second, direct subtraction is not valid. 

Example 3: Find the unit digit of 123 - 12356.

Unit digits: 3, 6.

Since 3 < 6, add 10 to 3: 3 + 10 = 13.

Subtract: 13 - 6 = 7.

The unit digit is 7.

Unit Digit Practice: Multiplication

Example 1: Find the unit digit of 7 × 8.

7 × 8 = 56. The unit digit is 6.

 Example 2: Find the unit digit of (Number ending in 6) × (Number ending in 6) × (Number ending in 7) × (Number ending in 3) × (Number ending in 1).

Unit digits: 6, 6, 7, 3, 1.

Multiply: (6 × 6 = 6); (7 × 3 = 1).

Final product of unit digits: 6 × 1 × 1 = 6.

The unit digit is 6.

Properties of Unit Digits with 5

1. 5 × Odd Number: If 5 is multiplied by any odd number, the unit digit will always be 5.

• Example: 5 × 7 = 35. Unit digit 5.

2. 5 × Even Number: If 5 is multiplied by any even number, the unit digit will always be 0.

• Example: 5 × 4 = 20. Unit digit 0.

Special Case: Guaranteeing a Zero Unit Digit in Products

If a series of multiplications includes numbers whose unit digits multiply to 0 (e.g., 2 × 5), then the unit digit of the entire product will be 0.

 Example: Product involving 82 (2) and 85 (5). Since 2 × 5 = 10, the unit digit of the product is 0.

Algebraic Unit Digit Problem

Example: (SSC CGL Tier 2 Question)

 Given: 3 × 6 × n, where 'n' is a single digit. If the unit digit of (3 × 6 × n) = (n + 2), find 'n'.

1. Unit digit of (3 × 6 × n) = Unit digit of (18 × n) = Unit digit of (8 × n).

2. Equation: Unit digit of (8 × n) = Unit digit of (n + 2).

3. Test n = 6:

• Unit digit of (8 × 6) = 48 -> 8.

• Unit digit of (6 + 2) = 8 -> 8.

4. Both sides are 8, so n = 6 is the solution.

Unit Digit with Powers: Cyclicity Rules

Cyclicity rules help find the unit digit of a number raised to a power.

Rule 1: No Change (0, 1, 5, 6)

If the last digit (unit digit) of a number is 0, 1, 5, or 6, its unit digit will not change regardless of the power (n).

• (1230)^12351 -> 0.

• (12131)^2356 -> 1.

• (12345)^n -> 5.

• (12356)^1235620 -> 6.

Rule 2: Cyclicity of 2 (4, 9)

For numbers ending in 4 or 9, the unit digit depends on whether the power (n) is odd or even. 

Example: Find the unit digit of (1234)^12305.

 Base unit digit 4. Power 12305 is odd. The unit digit is 4.

Rule 3: Cyclicity of 4 (2, 3, 7, 8)

For numbers ending in 2, 3, 7, or 8, follow these steps:

1. Take the last two digits of the power.

2. Divide these two digits by 4.

3. The remainder is the new power for the base's unit digit.

Example: Find the unit digit of (1237)^143.

Base unit digit 7. Power's last two digits 43.

43 ÷ 4 = 10, remainder 3.

New power for 7 is 3. 7^3 = 343. Unit digit is 3.

Rule 3 Special Case: Power is a Multiple of 4

If the remainder from dividing the last two digits of the power by 4 is 0, use 4 as the new power. (Memory Tip: If remainder = 0, then new power = 4.)

Example: Find the unit digit of (12378)^24.

Base unit digit 8. Power 24.

24 ÷ 4 = 6, remainder 0.

 New power for 8 is 4. 8^4 = 4096. Unit digit is 6.

Unit Digit with Factorial Powers

Unit Digit with Factorial Powers is an important concept in number systems where the exponent is a factorial (n!). By using unit digit cycles and factorial properties, you can quickly find the unit digit of large powers without performing lengthy calculations. 

Rule for Factorial Powers

When the power is a factorial:

• If the base's unit digit is 0, 1, 5, or 6, the unit digit remains unchanged.

• For other base unit digits (2, 3, 4, 7, 8, 9): If the factorial power is n! where n ≥ 4, this value is a multiple of 4.

• Thus, division by 4 yields a remainder of 0.

• The new power used for the base's unit digit will be 4.

Example 1: Find the unit digit of (Number ending in 2)^(X!) × (Number ending in 3)^(Y!) × (Number ending in 4)^(Z!), where X, Y, Z ≥ 4.

All powers are ≥ 4!, so use 4 as the effective power.

• 2^4: 16 -> 6.

• 3^4: 81 -> 1.

• 4^4: 256 -> 6.
  Multiply unit digits: 6 × 1 × 6 = 36.
  The unit digit is 6.

Example 2: Find the unit digit of 1!^1! + 2!^2! + 3!^3! + 4!^4! + 5!^5! + …

• 1!^1!: 1^1 = 1.

• 2!^2!: 2^2 = 4.

• 3!^3!: 6^6 -> 6.

• 4!^4!: 24^24. Base 4, power even. Unit digit 6.

• 5!^5!: Base 5! ends in 0. Unit digit 0.

• Subsequent terms: unit digit 0.
  Sum: 1 + 4 + 6 + 6 + 0 + … = 17.
  The unit digit is 7.

Important Rules of Unit Digit for SSC MTS Exam

The following rules help candidates find the unit digit of large numbers and powers within seconds:

• The unit digit is always the last digit of a number.

• In addition, subtraction and multiplication, only the unit digits affect the final unit digit.

• Numbers ending in 0, 1, 5, and 6 retain the same unit digit regardless of the power.

• Numbers ending in 4 and 9 follow an odd-even pattern for their unit digits.

• Numbers ending in 2, 3, 7, and 8 follow a cyclicity pattern of 4.

• If the power leaves a remainder of 0 when divided by 4, use 4 as the effective power.

• For factorial powers (n!, where n ≥ 4), use 4 as the effective power for bases ending in 2, 3, 4, 7, 8, and 9.

• Any factorial n! where n ≥ 5 has a unit digit of 0.

• Multiplication involving unit digits 2 and 5 guarantees a final unit digit of 0.

Memorizing cyclicity patterns can significantly reduce calculation time in SSC MTS Number System questions.

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