Number System, HCF and LCM Formula Guide for SSC Exams
Rahul Kumar
SSC Exam Expert & Content Editor
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The number system is the base of every Quantitative Aptitude question: HCF and LCM, remainders, divisibility, simplification and series all depend on it. This number system, HCF and LCM formula guide for SSC gives the types of numbers, the divisibility tests, the formulas for factors and series, the unit digit cycles and the HCF-LCM shortcuts, with solved examples.
Types of Numbers
| Type | Meaning | Example |
|---|---|---|
| Natural numbers | Counting numbers from 1 | 1, 2, 3, 4 |
| Whole numbers | Natural numbers and 0 | 0, 1, 2, 3 |
| Integers | Whole numbers and negative numbers | −2, 0, 5 |
| Prime numbers | Exactly two factors, 1 and itself | 2, 3, 5, 7, 11 |
| Composite numbers | More than two factors | 4, 6, 8, 9 |
| Co-prime numbers | Two numbers whose HCF is 1 | 8 and 15 |
| Rational numbers | Can be written as p/q | 1/2, 0.75 |
| Irrational numbers | Cannot be written as p/q | √2, π |
2 is the only even prime number, and 1 is neither prime nor composite.
Divisibility Rules
| Divisor | Test |
|---|---|
| 2 | The last digit is even |
| 3 | The sum of the digits is divisible by 3 |
| 4 | The last two digits are divisible by 4 |
| 5 | The last digit is 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 7 | Double the last digit and subtract it from the rest; the result must be divisible by 7 |
| 8 | The last three digits are divisible by 8 |
| 9 | The sum of the digits is divisible by 9 |
| 10 | The last digit is 0 |
| 11 | The difference between the sum of the digits at odd places and at even places is 0 or a multiple of 11 |
Factors
If a number N = ap × bq × cr, where a, b and c are primes, then:
| Item | Formula |
|---|---|
| Number of factors | (p + 1)(q + 1)(r + 1) |
| Sum of the factors | [(a^(p+1) − 1)/(a − 1)] × [(b^(q+1) − 1)/(b − 1)] × [(c^(r+1) − 1)/(c − 1)] |
| A perfect square | All the powers p, q and r are even; it has an odd number of factors |
Sum of Series
| Series | Sum |
|---|---|
| First n natural numbers | n(n + 1) / 2 |
| First n odd numbers | n² |
| First n even numbers | n(n + 1) |
| Squares of the first n natural numbers | n(n + 1)(2n + 1) / 6 |
| Cubes of the first n natural numbers | [n(n + 1) / 2]² |
Unit Digit Cycles
| Digit at the end of the number | Cycle of the unit digit |
|---|---|
| 0, 1, 5, 6 | Same digit always |
| 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 (or by 2 for the cycles of two). The remainder tells the place in the cycle, and the remainder 0 means the last digit of the cycle.
HCF and LCM Formulas
| Item | Formula or rule |
|---|---|
| Product of two numbers | HCF × LCM |
| HCF of fractions | HCF of the numerators / LCM of the denominators |
| LCM of fractions | LCM of the numerators / HCF of the denominators |
| Greatest number that divides a, b and c and leaves the same remainder r | HCF of (a − r), (b − r) and (c − r) |
| Greatest number that divides a, b and c leaving remainders r1, r2 and r3 | HCF of (a − r1), (b − r2) and (c − r3) |
| Smallest number divisible by a, b and c | LCM of a, b and c |
| Smallest number that leaves the same remainder r when divided by a, b and c | LCM + r |
| Smallest number that leaves a constant gap k (divisor minus remainder) in each case | LCM − k |
| Events that happen together again (bells, lights) | LCM of the time gaps |
Squares and Cubes to Remember
| Number | Square | Number | Square | Number | Cube |
|---|---|---|---|---|---|
| 11 | 121 | 19 | 361 | 2 | 8 |
| 12 | 144 | 20 | 400 | 3 | 27 |
| 13 | 169 | 21 | 441 | 4 | 64 |
| 14 | 196 | 22 | 484 | 5 | 125 |
| 15 | 225 | 23 | 529 | 6 | 216 |
| 16 | 256 | 24 | 576 | 7 | 343 |
| 17 | 289 | 25 | 625 | 8 | 512 |
| 18 | 324 | 9 | 729 |
Solved Examples
Example 1. Find the number of factors of 360.
360 = 2³ × 3² × 5. The number of factors is (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24.
Example 2. Find the unit digit of 745.
The cycle of 7 is 7, 9, 3, 1 (length 4). 45 divided by 4 leaves a remainder of 1, so the unit digit is the first one in the cycle: 7.
Example 3. Find the greatest number that divides 125 and 205 and leaves a remainder of 5 in each case.
HCF of (125 − 5) and (205 − 5) = HCF of 120 and 200 = 40.
Example 4. Find the smallest number which, when divided by 12, 15 and 20, leaves a remainder of 5 in each case.
LCM of 12, 15 and 20 = 60. The number is 60 + 5 = 65.
Example 5. The HCF of two numbers is 6 and their LCM is 36. If one number is 12, find the other.
The other number = (6 × 36) / 12 = 18.
Quick Tips
- Write a number as a product of primes before you use any factor formula.
- For unit digits, always find the remainder of the power by 4.
- Subtract the remainder first in the greatest-number problems, and add it last in the smallest-number problems.
- Learn the squares up to 25 and the cubes up to 10 by heart.
Quick Revision Points
- Number of factors = (p + 1)(q + 1)(r + 1).
- Sum of the first n natural numbers = n(n + 1)/2.
- HCF × LCM = product of the two numbers.
- Smallest number with the same remainder r = LCM + r.
- A number is divisible by 9 if its digit sum is divisible by 9.
Practice Questions With Answers
- Find the number of factors of 36.
Answer: 9 - Find the sum of the first 15 natural numbers.
Answer: 120 - 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 their HCF is 5. Find the LCM.
Answer: 120 - Find the sum of the first 10 odd numbers.
Answer: 100 - Is 1331 divisible by 11?
Answer: Yes, because 1 − 3 + 3 − 1 = 0 - Find the smallest number which leaves a remainder of 3 when divided by 8 and 12.
Answer: 27 - Find the square of 25.
Answer: 625
How to Practise This Chapter
Write each formula on a card, with a solved example on the back, and revise the cards every day. Then test yourself with our practice papers and the previous year papers page. See all guides on the Formula Guide page, and also read the Mathematics notes and Quantitative Aptitude notes. The previous guide is Physics and Chemistry and the next guide is Percentage, Profit and Loss, and Discount.
Frequently Asked Questions
What is the formula for the number of factors?
If N = a to the power p times b to the power q, then the number of factors is (p + 1)(q + 1).
What is the sum of the first n natural numbers?
It is n(n + 1)/2.
What is the sum of the first n odd numbers?
It is n squared.
What is the relation between HCF and LCM?
The product of two numbers is equal to the product of their HCF and LCM.
How do I find the unit digit of a large power?
Use the cycle of the unit digit and divide the power by the length of the cycle, which is 4 for 2, 3, 7 and 8.
What is the divisibility rule of 11?
The difference between the sum of the digits at odd places and at even places must be 0 or a multiple of 11.
How do I find the greatest number that leaves the same remainder?
Subtract the remainder from each number and find the HCF of the results.
How do I find the smallest number that leaves the same remainder?
Find the LCM of the divisors and add the remainder.
Which numbers have an odd number of factors?
Perfect squares have an odd number of factors.
Is 1 a prime number?
No. 1 is neither prime nor composite.
Is this topic asked in SSC exams?
Yes. Number system questions are asked in every SSC maths paper.
How many questions come from this topic in SSC exams?
Usually two to four questions, but the number changes from exam to exam.
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Physics and Chemistry Formula Guide for SSC Exams
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Percentage, Profit and Loss, and Discount Formula Guide for SSC Exams
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.