Remainders, Cyclicity and Unit Digit Tricks: Remainder Theorem Questions with Solved Examples for SSC
Rahul Kumar
SSC Exam Expert & Content Editor
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PDF Notes: Remainders, Cyclicity and Unit Digit Tricks: Remainder Theorem Questions with Solved Examples for SSC
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Questions on the unit digit of a power and the remainder of a big number look hard, but they can be solved in seconds with the idea of cyclicity and negative remainders. This guide on remainders, cyclicity and unit digit for SSC gives the cycles of all digits, the remainder tricks and many solved examples, with practice questions and FAQs.
Cyclicity of Unit Digits
The unit digit of powers of a number repeats in a cycle. To find the unit digit of a^n, find the cycle of the last digit of a and then find n mod the cycle length.
| Last digit of base | Cycle of unit digits | Cycle length |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 1 |
| 2 | 2, 4, 8, 6 | 4 |
| 3 | 3, 9, 7, 1 | 4 |
| 4 | 4, 6 | 2 |
| 5 | 5 | 1 |
| 6 | 6 | 1 |
| 7 | 7, 9, 3, 1 | 4 |
| 8 | 8, 4, 2, 6 | 4 |
| 9 | 9, 1 | 2 |
If the remainder on dividing the power by the cycle length is 0, take the last number of the cycle.
Remainder Basics
- Remainder of a sum = remainder of the sum of remainders.
- Remainder of a product = remainder of the product of remainders.
- A number can be replaced by its remainder (or its negative remainder) in a power.
- If a ≡ −1 (mod n), then a^k leaves remainder 1 for even k and n − 1 for odd k.
- Remainder of a number divided by 9 = remainder of the sum of its digits divided by 9.
Useful Remainder Cycles
| Case | Result |
|---|---|
| 2^n divided by 3 | Remainder 2 if n is odd, 1 if n is even |
| 2^n divided by 7 | Cycle 2, 4, 1 (period 3) |
| 3^n divided by 5 | Cycle 3, 4, 2, 1 (period 4) |
| 10^n divided by 9 | Always 1 |
| (n−1)^k divided by n | 1 if k even, n−1 if k odd |
Solved Examples
Example 1. Find the unit digit of 7^123.
Cycle of 7 is 4. 123 ÷ 4 leaves 3, so the third number: 7, 9, 3. Unit digit = 3.
Example 2. Find the unit digit of 2^50.
50 ÷ 4 leaves 2, so the second number of 2, 4, 8, 6 is 4.
Example 3. Find the unit digit of 3^99.
99 ÷ 4 leaves 3, so the third number of 3, 9, 7, 1 is 7.
Example 4. Find the remainder of 17^23 divided by 4.
17 leaves remainder 1 when divided by 4, so 1^23 = 1.
Example 5. Find the remainder of 2^100 divided by 7.
2^3 = 8 leaves 1. 100 = 3 × 33 + 1, so the remainder is 2^1 = 2.
Example 6. Find the remainder of 3^50 divided by 5.
3^4 = 81 leaves 1. 50 ÷ 4 leaves 2, so the remainder is 3^2 = 9, which leaves 4 when divided by 5.
Example 7. Find the unit digit of 1! + 2! + 3! + 4! + 5! + … + 10!.
From 5! onwards each term ends in 0. 1 + 2 + 6 + 24 = 33, so the unit digit is 3.
Example 8. Find the remainder of 15^25 divided by 16.
15 ≡ −1 and 25 is odd, so the remainder is 16 − 1 = 15.
Example 9. Find the unit digit of 13^13 × 17^17.
13^13: 3 with 13 ÷ 4 leaves 1, so 3. 17^17: 7 with 17 ÷ 4 leaves 1, so 7. 3 × 7 = 21, so the unit digit is 1.
Tricks
- Learn the cycle table by heart; digits 0, 1, 5 and 6 never change.
- For 4 and 9, check whether the power is odd or even.
- When the base is close to a multiple of the divisor, use the negative remainder method.
- For the unit digit of a product, multiply the unit digits of the factors.
- Divide the power by 4 for digits 2, 3, 7 and 8.
Quick Tips
- Memorise the cyclicity table for digits 2, 3, 7 and 8.
- Use negative remainders for bases close to the divisor.
- Use the cycle length 4 for 2, 3, 7, 8 and 2 for 4, 9.
- If the power leaves 0 on division by 4, take the last number of the cycle.
Quick Revision Points
- Unit digits of powers repeat in cycles of length 1, 2 or 4.
- Digits 0, 1, 5 and 6 keep the same unit digit for every power.
- Cycle of 2: 2, 4, 8, 6; of 3: 3, 9, 7, 1.
- a ≡ −1 (mod n) gives remainders 1 or n − 1.
- The unit digit of n! is 0 for n ≥ 5.
Practice Questions With Answers
- Find the unit digit of 2^2024.
Answer: 6 - Find the unit digit of 7^45.
Answer: 7 - Find the unit digit of 9^101.
Answer: 9 - Find the unit digit of 8^77.
Answer: 8 - Find the remainder when 5^30 is divided by 6.
Answer: 1 - Find the remainder when 2^50 is divided by 3.
Answer: 1 - Find the unit digit of 13^13 × 17^17.
Answer: 1 - Find the remainder when 15^25 is divided by 16.
Answer: 15 - Find the unit digit of 1! + 2! + 3! + 4! + 5!.
Answer: 3 - Find the remainder when 6^50 is divided by 7.
Answer: 1
How to Practise This Chapter
Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Quantitative Aptitude notes page, read the previous topic Races and Circular Tracks.
Frequently Asked Questions
What is cyclicity?
It is the repeating pattern of the last digits of the powers of a number.
What is the cycle of the unit digit of 2?
2, 4, 8, 6, with a length of 4.
What is the cycle of 7?
7, 9, 3, 1, with a length of 4.
Which digits do not change in powers?
The digits 0, 1, 5 and 6.
How do I find the unit digit of a^n?
Find the cycle of the last digit of a, divide n by the cycle length and pick the number at the position of the remainder.
What if the remainder is 0 when dividing the power by 4?
Take the last number of the cycle.
What is the negative remainder method?
Replace the base by its negative remainder, for example 15 by −1 when dividing by 16, and then raise it to the power.
What is the remainder when 10^n is divided by 9?
It is always 1.
How do I find the unit digit of a product?
Multiply the unit digits of the factors and take the last digit.
What is the unit digit of n! for n greater than 4?
It is 0, because 5! and above contain a factor of 10.
Are these topics asked in SSC?
Yes. Unit digit and remainder questions are common in SSC CGL and CHSL.
How much time should a question take?
About 30 seconds.
Rahul Kumar
SSC Exam Expert & Content Editor
Rahul is a senior SSC exam strategist with 8+ years of experience helping aspirants crack CGL, CHSL and MTS exams. He writes in-depth notification breakdowns, exam pattern guides and preparation strategies.