Number System Tricks: Divisibility, Remainder, Unit Digit, HCF and LCM Questions for SSC CGL, CHSL
Rahul Kumar
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Number system is the base chapter of SSC maths. It gives direct questions on divisibility, remainders, unit digits, factors, HCF and LCM, and it supports simplification and data interpretation. These number system tricks for SSC explain the divisibility rules, the remainder and unit digit methods and the HCF-LCM shortcuts, with solved questions.
Divisibility Rules
| By | Test |
|---|---|
| 2 | Last digit is even |
| 3 | Sum of the digits is divisible by 3 |
| 4 | Last two digits are divisible by 4 |
| 5 | Last digit is 0 or 5 |
| 6 | Divisible by 2 and 3 |
| 8 | Last three digits are divisible by 8 |
| 9 | Sum of the digits is divisible by 9 |
| 11 | Difference of the sums of alternate digits is 0 or a multiple of 11 |
Unit Digit Cycles
| Number ends in | Cycle of the unit digit |
|---|---|
| 0, 1, 5, 6 | Same digit |
| 2 | 2, 4, 8, 6 |
| 3 | 3, 9, 7, 1 |
| 4 | 4, 6 |
| 7 | 7, 9, 3, 1 |
| 8 | 8, 4, 2, 6 |
| 9 | 9, 1 |
To find the unit digit of a power, divide the power by 4 and use the remainder (0 means the last digit of the cycle).
Factors, HCF and LCM
| Item | Rule |
|---|---|
| Number of factors of a^p × b^q | (p + 1)(q + 1) |
| HCF × LCM | Product of the two numbers |
| Greatest number that divides a and b and leaves the same remainder r | HCF of (a − r) and (b − r) |
| Smallest number that leaves remainder r when divided by a, b and c | LCM + r |
Solved Examples
Example 1. Is 1,234,563 divisible by 3?
Digit sum = 1 + 2 + 3 + 4 + 5 + 6 + 3 = 24, which is divisible by 3. So yes.
Example 2. Find the unit digit of 745.
The cycle of 7 has length 4. 45 ÷ 4 leaves 1, so the digit is the first in the cycle: 7.
Example 3. Find the number of factors of 360.
360 = 2³ × 3² × 5. Factors = 4 × 3 × 2 = 24.
Example 4. Find the greatest number that divides 125 and 205 and leaves a remainder of 5.
HCF of 120 and 200 = 40.
Example 5. Find the smallest number that leaves a remainder of 5 when divided by 12, 15 and 20.
LCM = 60, so the number = 60 + 5 = 65.
Example 6 (remainder). What is the remainder when 210 is divided by 7?
2³ = 8 leaves 1. So 29 leaves 1, and 210 = 29 × 2 leaves 2.
Example 7. The HCF of two numbers is 6 and the LCM is 36. One number is 12. Find the other.
6 × 36/12 = 18.
Quick Tips
- Learn the divisibility rules of 2 to 11.
- For unit digits, divide the power by 4.
- Write the number in prime factors before you count factors.
- Subtract the remainder first for HCF and add it last for LCM.
Quick Revision Points
- A number is divisible by 9 if its digit sum is divisible by 9.
- Unit digit cycles have the length 4, 2 or 1.
- HCF × LCM = product of the two numbers.
- Number of factors = (p + 1)(q + 1)(r + 1).
- LCM + r gives the same remainder r.
Practice Questions With Answers
- Find the number of factors of 36.
Answer: 9 - Find the unit digit of 322.
Answer: 9 - Find the LCM of 6 and 8.
Answer: 24 - Find the HCF of 36 and 48.
Answer: 12 - The product of two numbers is 600 and the HCF is 5. Find the LCM.
Answer: 120 - Is 1331 divisible by 11?
Answer: Yes (1 − 3 + 3 − 1 = 0) - Find the smallest number that leaves 3 when divided by 8 and 12.
Answer: 27 - Find the remainder when 210 is divided by 7.
Answer: 2 - Find the sum of the first 10 odd numbers.
Answer: 100 - What is the unit digit of 210?
Answer: 4
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 Mathematics notes page, read the previous topic Time, Speed and Distance Tricks and the next topic Trigonometry Tricks. Formulas are in the Formula Guide and more practice is in the Quantitative Aptitude notes. Also read the Number System, HCF and LCM Formula Guide.
Frequently Asked Questions
What is the divisibility rule of 9?
A number is divisible by 9 if the sum of its digits is divisible by 9.
What is the divisibility rule of 11?
The difference of the sum of the digits at odd and even places is 0 or a multiple of 11.
How do I find the unit digit of a large power?
Use the cycle of the unit digit and the remainder of the power divided by 4.
How many factors does a number have?
Write it as a product of primes and multiply (power + 1) for each prime.
What is the relation between HCF and LCM?
The product of two numbers equals HCF times LCM.
What is the greatest number that leaves the same remainder?
The HCF of the differences after subtracting the remainder from each number.
What is the smallest number that leaves the same remainder?
LCM of the divisors plus the remainder.
Is 1 a prime number?
No. It is neither prime nor composite.
Is number system asked in SSC exams?
Yes. It is asked in every SSC maths paper.
What are the common mistakes?
Forgetting to add the remainder at the end and wrong cycles of the unit digits.
How much time should I take?
About 30 to 40 seconds.
Which numbers have an odd number of factors?
Perfect squares.
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Trigonometry Tricks: Ratios, Identities, Height and Distance Questions for SSC CGL, CHSL
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.