Number System — RRB NTPC Mathematics
Number System carries roughly 8% of Mathematics's 30 questions (see
docs/exam-briefs/rrb-ntpc-2026-brief.mdfor CBT 1's overall structure — 100 questions, 90 minutes, +1/−1/3 marking). Every question type in this chapter can be solved in under 30 seconds once the underlying rule is automatic.
1. What RRB NTPC actually asks
Number System questions in CBT 1 typically test: place value versus face value, classification of numbers (natural, whole, integer, rational), divisibility rules, identifying prime numbers, unit-digit cyclicity of powers, basic remainder problems, and BODMAS-based simplification. Nothing here demands more than a clean, memorised rule applied carefully — the difficulty is almost entirely in avoiding a careless slip under time pressure, not in the underlying concept.
2. Place value and face value
The face value of a digit is the digit itself, regardless of position. The place value is the face value multiplied by the value of the position it occupies (units, tens, hundreds, and so on).
In the number 6,472: the face value of 7 is 7; the place value of 7 is 7 × 10 = 70. Face value and place value are identical only for a digit in the units place.
3. Number classification
| Set | Definition | Examples |
|---|---|---|
| Natural numbers | Counting numbers starting from 1 | 1, 2, 3, ... |
| Whole numbers | Natural numbers plus zero | 0, 1, 2, 3, ... |
| Integers | Whole numbers plus negatives | ..., −2, −1, 0, 1, 2, ... |
| Rational numbers | Expressible as p/q, q ≠ 0 | 1/2, −3, 0.75 |
| Irrational numbers | Not expressible as p/q | √2, π |
| Prime numbers | Exactly two distinct factors: 1 and itself | 2, 3, 5, 7, 11 |
| Composite numbers | More than two factors | 4, 6, 8, 9 |
2 is the only even prime number — every other even number is divisible by 2 and therefore has at least three factors (1, 2, itself). 1 is neither prime nor composite, since a prime requires exactly two distinct factors and 1 has only one.
4. Divisibility rules
| Divisor | Rule |
|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) |
| 3 | Sum of digits divisible by 3 |
| 4 | Last two digits form a number divisible by 4 |
| 5 | Last digit is 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 8 | Last three digits form a number divisible by 8 |
| 9 | Sum of digits divisible by 9 |
| 11 | Difference between the sum of digits in odd positions and even positions is 0 or divisible by 11 |
These rules answer "is X divisible by Y" without performing the division — the entire point is speed, so they are worth memorising cold rather than re-derived.
5. Unit digit cyclicity
The unit digit of any power follows a repeating cycle of at most 4. To find the unit digit of a^b: find the unit digit's own cycle (compute a^1, a^2, a^3, a^4 and note the unit digits repeat from there), then take b mod 4 (using 4 instead of 0) to pick the position in that cycle.
For example, powers of 7 cycle through unit digits 7, 9, 3, 1 (for exponents 1, 2, 3, 4) and then repeat. To find the unit digit of 7^123: 123 mod 4 = 3, so the answer is the third value in the cycle, which is 3.
6. Simplification (BODMAS)
Evaluate in this strict order: Brackets, Of, Division, Multiplication, Addition, Subtraction. Division and multiplication are evaluated left to right as they appear (division does not automatically come before multiplication if multiplication appears first in the expression), and the same applies to addition and subtraction.
Worked examples
Q1. What is the unit digit of 7¹²³?
Pick an option to check your answer.
Show explanation
Solution. Powers of 7 cycle through unit digits 7, 9, 3, 1 for exponents 1, 2, 3, 4, then repeat. 123 mod 4 = 3, so the unit digit matches the 3rd position in the cycle, which is 3.
(a) would be correct only if 123 mod 4 were 0 (i.e., exponent a multiple of 4). (c) is the unit digit for exponent 1 or 5, not 123. (d) is the unit digit for exponent 2 or 6. Answer: (b).
Q2. Simplify: 45 + [10 − {6 − (5 − 3)}]
Pick an option to check your answer.
Show explanation
Solution. Work from the innermost bracket outward. (5 − 3) = 2. Then {6 − 2} = 4. Then [10 − 4] = 6. Finally, 45 + 6 = 51.
A common error is resolving brackets in the wrong order or misreading which bracket nests inside which — always start from the innermost parentheses and work outward, one layer at a time. Answer: (c).
8. Common traps
- Confusing face value with place value. Face value never changes with position; place value depends entirely on it.
- Treating 1 as prime. A prime number must have exactly two distinct factors — 1 has only one, so it is excluded by definition.
- Getting the unit-digit cycle position wrong when the exponent is an exact multiple of 4. Use 4, not 0, as the cycle position in that case (e.g., for exponent 4, 8, 12, use position 4 of the cycle, not position 0).
- Resolving nested brackets out of order. Always start from the innermost bracket and work outward.
- Applying division before multiplication automatically. Both are evaluated strictly left to right in the order they appear in the expression, not division-always-first.
9. Guessing strategy
With RRB NTPC's 1/3 negative marking and zero penalty for skipping (see docs/exam-briefs/rrb-ntpc-2026-brief.md), a blind guess among 4 options has an expected value of −0.083 marks — worse than skipping. Only guess once you've eliminated at least one option, which brings the odds to break-even or better; a divisibility check or a quick parity check is often enough to eliminate one or two options in seconds even without solving the question fully.
Summary
- Number System is roughly 8% of Mathematics's 30 questions in CBT 1.
- Face value is the digit itself; place value is face value × positional weight — identical only in the units place.
- 2 is the only even prime; 1 is neither prime nor composite.
- Divisibility rules for 2, 3, 4, 5, 6, 8, 9, and 11 should be automatic recall, not re-derived.
- Unit-digit cyclicity repeats every 4 powers — use the exponent mod 4 (treating a result of 0 as position 4) to find the position.
- BODMAS order is strict, but division/multiplication and addition/subtraction within their tiers go strictly left to right.
- A blind 4-option guess has negative expected value under RRB NTPC's marking scheme — only guess after eliminating at least one option.
