Number Theory: Divisibility, Primes, Factors, Remainders and Trailing Zeros for SSC
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Number theory is the base of many SSC maths questions: divisibility, prime numbers, factors, HCF and LCM, remainders and unit digits. A few formulas and rules are enough to answer most of them quickly. This note gives all of them with solved examples, practice questions and FAQs.
Divisibility Rules
| Divisor | Rule |
|---|---|
| 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 both 2 and 3 |
| 8 | Last three digits are divisible by 8 |
| 9 | Sum of the digits is divisible by 9 |
| 10 | Last digit is 0 |
| 11 | The difference between the sum of digits at odd places and at even places is 0 or a multiple of 11 |
| 7 | Double the last digit and subtract it from the rest; repeat; the result should be 0 or a multiple of 7 |
| 12 | Divisible by both 3 and 4 |
| 15 | Divisible by both 3 and 5 |
Prime Numbers
- A prime number has exactly two factors: 1 and itself. 1 is neither prime nor composite.
- 2 is the only even prime number.
- There are 25 primes below 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
- There are 15 primes below 50 and 10 primes below 30.
- To test whether n is prime, check divisibility by primes up to √n.
- Co-prime numbers have HCF = 1 (they need not be prime themselves, for example 8 and 15).
- Twin primes differ by 2: (3, 5), (5, 7), (11, 13), (17, 19).
Factors of a Number
If N = ap × bq × cr (a, b, c are primes), then:
| Quantity | Formula |
|---|---|
| Number of factors | (p + 1)(q + 1)(r + 1) |
| Sum of factors | [(ap+1 − 1)/(a − 1)] × [(bq+1 − 1)/(b − 1)] × … |
| Product of all factors | N(number of factors)/2 |
| Number of ways to write N as a product of two factors | Half of the number of factors (for a non-perfect square) |
| A number has an odd number of factors | Only if it is a perfect square |
HCF and LCM
- HCF × LCM = product of the two numbers.
- The HCF of a, b divides both; the LCM is divisible by both.
- For fractions: HCF = HCF of numerators / LCM of denominators; LCM = LCM of numerators / HCF of denominators.
- The smallest number divisible by all numbers from 1 to 10 is 2520.
- For the largest number that divides a, b and c leaving the same remainder r, find the HCF of (a − r), (b − r), (c − r).
Trailing Zeros in n!
Number of trailing zeros = ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + … (count the power of 5).
Unit Digit Cycles
| Digit | Cycle of the unit digit | Length |
|---|---|---|
| 2 | 2, 4, 8, 6 | 4 |
| 3 | 3, 9, 7, 1 | 4 |
| 7 | 7, 9, 3, 1 | 4 |
| 8 | 8, 4, 2, 6 | 4 |
| 4 | 4, 6 | 2 |
| 9 | 9, 1 | 2 |
| 0, 1, 5, 6 | Same digit | 1 |
For the unit digit of an, divide n by the cycle length and use the remainder (if the remainder is 0, take the last digit of the cycle).
Remainder Facts
- Remainders are added, subtracted and multiplied in the same way as the numbers (the remainder of the sum is the sum of the remainders, reduced).
- Fermat’s little theorem: if p is prime and a is not a multiple of p, then ap−1 leaves remainder 1 when divided by p.
- Wilson’s theorem: (p − 1)! + 1 is divisible by a prime p.
- The remainder when an is divided by (a + 1) is 1 if n is even and a if n is odd.
- Sum of the first n natural numbers = n(n + 1)/2; of odd numbers = n²; of squares = n(n + 1)(2n + 1)/6; of cubes = [n(n + 1)/2]².
Solved Examples
| Question | Solution |
|---|---|
| Number of factors of 360 | 360 = 2³ × 3² × 5, so (3 + 1)(2 + 1)(1 + 1) = 24 |
| Sum of the factors of 12 | 12 = 2² × 3; (2³ − 1)/1 × (3² − 1)/2 = 7 × 4 = 28 |
| Is 4356 divisible by 11? | (4 + 5) − (3 + 6) = 0, so yes |
| Trailing zeros in 100! | 20 + 4 = 24 |
| Unit digit of 745 | Cycle length 4; 45 mod 4 = 1, so the first digit of the cycle = 7 |
| Remainder when 210 is divided by 7 | 2³ = 8 ≡ 1 (mod 7); 10 = 3 × 3 + 1, so remainder = 2 |
| HCF of two numbers is 6 and their LCM is 60. One number is 12. Find the other. | Product = 360, so 360/12 = 30 |
| Primes between 1 and 30 | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 = 10 primes |
Quick Tips
- Write the prime factorisation first; most factor questions follow from it.
- For the unit digit, find the cycle and use the remainder of the power.
- For the remainder with large numbers, reduce the numbers first.
- Check if the number is a perfect square when asked about odd factors.
Quick Revision Points
- Number of factors = (p + 1)(q + 1)(r + 1).
- HCF × LCM = product of the two numbers.
- Trailing zeros in n! = ⌊n/5⌋ + ⌊n/25⌋ + ….
- 2 is the only even prime; 1 is not a prime.
- Unit digits of 2, 3, 7 and 8 have a cycle of 4.
Practice Questions With Answers
- How many factors does 36 have?
Answer: 9 - Which is the only even prime number?
Answer: 2 - Trailing zeros in 25!?
Answer: 6 - Unit digit of 34?
Answer: 1 - Is 1331 divisible by 11?
Answer: Yes - The LCM of 12 and 18?
Answer: 36 - The number of primes less than 20?
Answer: 8 - Sum of the first 10 natural numbers?
Answer: 55 - Which numbers have an odd number of factors?
Answer: Perfect squares - The smallest number divisible by 1 to 10?
Answer: 2520
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 Number Theory notes page.
Frequently Asked Questions
What is a prime number?
A number with exactly two factors, 1 and itself.
Is 1 a prime number?
No, 1 is neither prime nor composite.
How do I find the number of factors?
Write the prime factorisation and multiply the exponents plus one.
What is the relation between HCF and LCM?
HCF × LCM = product of the two numbers.
What is the rule for divisibility by 11?
The difference between the sum of the digits at odd and even places must be 0 or a multiple of 11.
How do I count the trailing zeros in n factorial?
Add n/5, n/25, n/125 and so on, taking the whole-number part each time.
How do I find a unit digit with a large power?
Use the cycle of the unit digit and the remainder of the power divided by the cycle length.
What are co-prime numbers?
Numbers whose HCF is 1.
What is Fermat’s little theorem?
If p is prime and a is not divisible by p, then a^(p−1) leaves remainder 1 on division by p.
Which numbers have an odd number of factors?
Perfect squares.
How many primes are there below 100?
25.
Is this topic asked in SSC?
Yes, number system questions appear in every SSC CGL, CHSL and CPO paper.
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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.