In this guide
Number system questions reward a particular kind of thinking: seeing the structure of a number instead of calculating blindly. No one expects you to compute 7^95. You are expected to notice that the unit digits of powers of 7 repeat in a cycle of four. Learn the handful of rules in this post and most number system questions become quick, reliable marks.
Types of numbers
| Type | Meaning | Examples |
|---|---|---|
| Natural numbers | Counting numbers | 1, 2, 3… |
| Whole numbers | Natural numbers and zero | 0, 1, 2… |
| Integers | Whole numbers and negatives | …−2, −1, 0, 1… |
| Rational | Can be written as p/q (q ≠ 0) | 3/4, 0.25, 0.333… |
| Irrational | Cannot be written as p/q | √2, π |
| Prime | Exactly two factors | 2, 3, 5, 7, 11… |
| Co-prime | HCF of the pair is 1 | 8 and 15 |
Divisibility rules
| Divisor | Rule |
|---|---|
| 2 | Last digit even |
| 3 | Sum of digits divisible by 3 |
| 4 | Last two digits divisible by 4 |
| 5 | Last digit 0 or 5 |
| 6 | Divisible by 2 and 3 |
| 8 | Last three digits divisible by 8 |
| 9 | Sum of digits divisible by 9 |
| 11 | Difference between sums of alternate digits is 0 or divisible by 11 |
Example: Is 5,72,396 divisible by 11? Alternate sums: (5 + 2 + 9) = 16 and (7 + 3 + 6) = 16. The difference is 0, so yes.
HCF and LCM
- HCF — the largest number that divides all given numbers.
- LCM — the smallest number divisible by all given numbers.
- For two numbers, HCF × LCM = product of the numbers.
Standard problem types
- Largest number that divides a, b and c leaving the same remainder → HCF of the differences (b − a), (c − b), (c − a).
- Largest number that divides a, b, c leaving remainders r1, r2, r3 → HCF of (a − r1), (b − r2), (c − r3).
- Smallest number which, when divided by a, b, c, leaves remainder r in each case → LCM(a, b, c) + r.
- Bells ringing together / runners meeting → LCM of the intervals.
Worked example: Find the largest number that divides 62, 132 and 237 leaving the same remainder.
Differences: 132 − 62 = 70, 237 − 132 = 105, 237 − 62 = 175. HCF of 70, 105 and 175 is 35.
Remainders
- The remainder of a sum or product equals the remainder of the sum or product of the individual remainders.
- Divisibility by 9: the remainder when a number is divided by 9 equals the remainder of its digit sum.
Worked example: Remainder when 47 × 53 is divided by 5?
47 leaves 2, 53 leaves 3; 2 × 3 = 6, which leaves 1 when divided by 5.
Unit digits
Powers repeat their unit digits in cycles:
| Base ends in | Cycle | 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 |
Worked example: Unit digit of 7^95. Divide 95 by 4: remainder 3. The third term in the cycle for 7 is 3. So the unit digit is 3.
Factors
If N = a^p × b^q × c^r (prime factorisation), then:
- number of factors = (p + 1)(q + 1)(r + 1);
- sum of factors = [(a^(p+1) − 1)/(a − 1)] × [(b^(q+1) − 1)/(b − 1)] × …
Worked example: Number of factors of 360.
360 = 2³ × 3² × 5¹. Factors = (3 + 1)(2 + 1)(1 + 1) = 24.
Trailing zeros in a factorial
The number of zeros at the end of n! equals the number of times 5 is a factor:
⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + …
Worked example: Trailing zeros in 100! = 20 + 4 = 24.
Practice
- Is 7,34,568 divisible by 8?
- Find the smallest number which, when divided by 12, 15 and 20, leaves a remainder of 5 in each case.
- Find the unit digit of 3^58.
- How many factors does 180 have?
- How many trailing zeros does 250! have?
- The HCF of two numbers is 12 and their product is 2,160. Find their LCM.
Answers: 1. Yes (568 ÷ 8 = 71). 2. 65. 3. 9 (58 leaves remainder 2 → second term of 3, 9, 7, 1). 4. 18 (2² × 3² × 5). 5. 62 (50 + 10 + 2). 6. 180.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Staff Selection Commission website .
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