In this guide
Number system questions test whether you know how numbers behave. They are among the shortest questions in the maths section: no long working, no formulas to rearrange. Once the rules are clear, most take well under a minute, which frees time for the longer arithmetic questions.
In RRB Group D these usually appear as divisibility checks, missing-digit questions, unit digits of large powers, counting factors, and simple remainder or sum questions.
Types of numbers
| Type | Meaning | Examples |
|---|---|---|
| Natural numbers | Counting numbers | 1, 2, 3, … |
| Whole numbers | Natural numbers and 0 | 0, 1, 2, … |
| Integers | Whole numbers and their negatives | …, −2, −1, 0, 1, 2, … |
| Rational numbers | Can be written as p/q, q ≠ 0 | 3/4, −5, 0.25 |
| Irrational numbers | Cannot be written as p/q | √2, π |
| Prime numbers | Exactly two factors, 1 and itself | 2, 3, 5, 7, 11, 13 |
| Composite numbers | More than two factors | 4, 6, 8, 9, 10 |
| Co-primes | Two numbers whose HCF is 1 | 8 and 15 |
The primes below 100 are: 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.
To check whether a number is prime, divide it only by the primes up to its square root. If none divides it, it is prime. For 91: √91 is a little under 10, so try 2, 3, 5, 7. Since 91 = 7 × 13, it is not prime.
Divisibility rules
| Divisible by | Rule | Example |
|---|---|---|
| 2 | Last digit is even | 3,458 |
| 3 | Sum of digits divisible by 3 | 7,293 (sum 21) |
| 4 | Last two digits divisible by 4 | 5,316 (16) |
| 5 | Last digit 0 or 5 | 8,745 |
| 6 | Divisible by both 2 and 3 | 4,122 |
| 8 | Last three digits divisible by 8 | 3,512 (512 = 8 × 64) |
| 9 | Sum of digits divisible by 9 | 5,742 (sum 18) |
| 10 | Last digit 0 | 6,530 |
| 11 | Sum of digits in odd places minus sum in even places is 0 or divisible by 11 | 4,356 |
For 11, count places from the right: the last digit is place 1.
Unit digits of powers
The last digit of a power repeats in a short cycle.
| Base ends in | Cycle of last digits | Cycle length |
|---|---|---|
| 0, 1, 5, 6 | Stays the same | 1 |
| 4 | 4, 6 | 2 |
| 9 | 9, 1 | 2 |
| 2 | 2, 4, 8, 6 | 4 |
| 3 | 3, 9, 7, 1 | 4 |
| 7 | 7, 9, 3, 1 | 4 |
| 8 | 8, 4, 2, 6 | 4 |
Method: divide the power by 4. The remainder gives the place in the cycle. A remainder of 0 means the last digit of the cycle.
Counting factors
Write the number as a product of primes: N = aᵐ × bⁿ × … Then the number of factors is (m + 1)(n + 1)…
Remainders
Dividend = divisor × quotient + remainder. The remainder is always less than the divisor.
Useful sums
- 1 + 2 + … + n = n(n + 1) ÷ 2. So 1 + 2 + … + 50 = 50 × 51 ÷ 2 = 1,275.
- Sum of the first n odd numbers = n².
- Sum of the first n even numbers = n(n + 1).
Worked examples
Example 1: Is 4,356 divisible by 11?
- Places from the right: 6 (1st), 5 (2nd), 3 (3rd), 4 (4th).
- Odd places: 6 + 3 = 9. Even places: 5 + 4 = 9.
- Difference = 0, so yes. Check: 11 × 396 = 4,356.
Example 2: Find the unit digit of 2¹⁵.
- 15 ÷ 4 leaves remainder 3.
- The third digit in the cycle 2, 4, 8, 6 is 8. Check: 2¹⁵ = 32,768.
Example 3: How many factors does 60 have?
- 60 = 2² × 3¹ × 5¹.
- Factors = (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12.
Example 4: Find the smallest three-digit number divisible by 13.
- 100 ÷ 13 = 7, remainder 9. So 100 is not a multiple.
- The next multiple is 13 × 8 = 104.
Example 5: 57x2 is divisible by 9. Find the digit x.
- Digit sum = 5 + 7 + x + 2 = 14 + x.
- The next multiple of 9 after 14 is 18, so x = 4. (27 would need x = 13, which is not a digit.)
- Check: 5,742 ÷ 9 = 638.
Example 6: In a division, the divisor is 17, the quotient is 9 and the remainder is 5. Find the dividend.
- Dividend = 17 × 9 + 5 = 153 + 5 = 158.
Common mistakes
- Using a digit-sum rule for 4 or 8. Those rules look at the last two or three digits, not the sum of digits.
- Forgetting that remainder 0 means the last digit of the unit-digit cycle, not the first.
- Calling 1 a prime number, or forgetting that 2 is prime.
- Counting factors by listing them one by one for a large number. Use the prime-power formula.
Practice set
- Find the unit digit of 3¹⁰.
- How many factors does 48 have?
- Is 7,293 divisible by 3?
- What is the largest two-digit number divisible by 7?
- Find the sum of the first 10 even numbers.
- Find the unit digit of 7⁴³ × 3²².
- What is the smallest number that must be added to 1,000 to make it divisible by 7?
- Find the sum of the first 15 odd numbers.
Answers:
- 9. 10 ÷ 4 leaves remainder 2; the second digit of 3, 9, 7, 1 is 9.
- 10. 48 = 2⁴ × 3; (4 + 1)(1 + 1) = 10.
- Yes. 7 + 2 + 9 + 3 = 21, which is divisible by 3.
- 98. 7 × 14 = 98; 7 × 15 = 105 has three digits.
- 110. n(n + 1) = 10 × 11.
- 7. 43 ÷ 4 leaves 3, so 7⁴³ ends in 3. 22 ÷ 4 leaves 2, so 3²² ends in 9. 3 × 9 = 27, which ends in 7.
- 1. 1,000 ÷ 7 = 142, remainder 6. 7 − 6 = 1, and 1,001 = 7 × 143.
- 225. n² = 15².
What to do next
- Write the unit-digit cycle table from memory until you can do it without a mistake.
- Learn the 25 primes below 100.
- Solve 15 mixed number system questions from past papers with a timer.
- Move on to LCM and HCF, which uses the same prime factors, and then BODMAS.
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 Railway Recruitment Boards website .
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