Why this topic matters · 8 min read
Number System is one of the highest-weightage topics in RRB NTPC Quant, appearing in almost every paper with 3-5 direct questions. Questions test divisibility rules, types of numbers, LCM/HCF concepts, remainders, unit digits, and properties of integers. Getting these basics right also helps in other topics like Time and Work, Fractions, and Simplification. Expect 1-2 tricky remainder or unit-digit questions alongside straightforward classification questions.
Types of Numbers
Numbers are classified in layers. Natural numbers start from 1, whole numbers from 0, integers include negatives too. Rational numbers can be written as p/q (q not zero), irrational numbers cannot (like root 2, pi). Real numbers include both. Prime numbers have exactly two factors: 1 and themselves. Composite numbers have more than two factors. 1 is neither prime nor composite — this is a favourite trap in RRB NTPC.
- Natural Numbers: 1, 2, 3, ... (no zero, no negatives)
- Whole Numbers: 0, 1, 2, 3, ... (zero included)
- Integers: ..., -2, -1, 0, 1, 2, ... (negatives included)
- Prime: exactly 2 factors. Only even prime is 2.
- 1 is neither prime nor composite — memorise this cold.
- Co-prime pairs share no common factor except 1. Example: 8 and 9.
Divisibility Rules (Speed Shortcuts)
Divisibility rules let you avoid actual division in the exam. RRB NTPC regularly asks: 'Which of these numbers is divisible by 11?' or 'Find the smallest number divisible by both X and Y.' Master rules for 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11. The rule for 11 is the most exam-favourite and most forgotten.
- Div by 2: last digit even. Div by 5: last digit 0 or 5.
- Div by 3: sum of digits divisible by 3. Div by 9: sum of digits divisible by 9.
- Div by 4: last TWO digits divisible by 4. Div by 8: last THREE digits divisible by 8.
- Div by 6: divisible by both 2 and 3.
- Div by 11: (sum of digits at odd positions) minus (sum of digits at even positions) = 0 or multiple of 11.
- Div by 7: double last digit, subtract from rest; result divisible by 7. (Rarely asked but possible.)
Key formulas
Rule for 11
(Sum of odd-position digits) - (Sum of even-position digits) = 0 or ±11, ±22...
When: When asked which number is divisible by 11, or to find missing digit in a number divisible by 11.
Worked examples
Is 85437 divisible by 11? Odd positions (1st, 3rd, 5th): 8+4+7=19. Even positions (2nd, 4th): 5+3=8. Difference: 19-8=11. Yes, divisible by 11.
Is 2376 divisible by 8? Last three digits: 376. 376 divided by 8 = 47. Exactly divisible. Yes.
Unit Digit (Last Digit) Patterns
RRB NTPC frequently asks: 'Find the unit digit of 7 raised to 55' or 'unit digit of a product of large numbers.' Every number's power cycles its unit digit in a pattern. You only need to find the remainder when the power is divided by the cycle length, then pick from the cycle.
- Unit digit of powers of 2: cycle is 2,4,8,6 — length 4.
- Unit digit of powers of 3: cycle is 3,9,7,1 — length 4.
- Unit digit of powers of 7: cycle is 7,9,3,1 — length 4.
- Unit digit of powers of 8: cycle is 8,4,2,6 — length 4.
- Digits 0,1,5,6 always end in themselves regardless of power.
- Digits 4 and 9 have cycle of 2: 4 gives 4,6,4,6... and 9 gives 9,1,9,1...
Key formulas
Unit Digit Step
Divide power by cycle length → use remainder to pick position in cycle. If remainder is 0, pick last element.
When: Any question asking unit digit of a^n for large n.
Worked examples
Unit digit of 7 to the power 55: Cycle of 7 is (7,9,3,1), length 4. 55 divided by 4 gives remainder 3. Third element in cycle is 3. Answer: 3.
Unit digit of 4 to the power 18: Cycle of 4 is (4,6), length 2. 18 divided by 2 = remainder 0. Take last element: 6. Answer: 6.
Remainders and Divisibility
Remainder questions appear in 2 forms: direct (what is remainder when X is divided by Y) and reverse (find the number that gives remainder R when divided by D). A key shortcut: if a number N gives remainder R when divided by D, then N = D times quotient plus R. For consecutive remainders involving multiple divisors, use LCM as the base.
- N = Divisor x Quotient + Remainder. This is the master formula.
- If N leaves same remainder R when divided by a, b, c — then (N - R) is divisible by LCM(a, b, c).
- If N leaves remainders that are each (Divisor - k) — meaning it is k short each time — then N + k is divisible by LCM.
- Remainder of a product: find remainder of each factor separately, multiply those remainders, then find remainder of that product.
- Negative remainder trick: remainder of 47 divided by 5 = -3 is same as +2 (since 5-3=2). Use whichever is easier.
Key formulas
Master Remainder
N = D * Q + R, where 0 <= R < D
When: Setting up any remainder problem.
Common Remainder (Same R)
N = LCM(a, b, c) * k + R
When: N leaves same remainder R when divided by a, b, and c.
Short by k (Complementary)
N = LCM(a, b, c) * k - constant_shortfall
When: N is always k less than a perfect multiple of each divisor.
Worked examples
Find the smallest number that when divided by 4, 6, and 8 leaves remainder 3 each time. LCM(4,6,8) = 24. Answer = 24 + 3 = 27.
Find smallest number that when divided by 5, 6, 7 leaves remainders 4, 5, 6 (i.e., always 1 short). LCM(5,6,7) = 210. Answer = 210 - 1 = 209.
Factors, HCF and LCM Quick Concepts
HCF is the greatest common divisor — useful when splitting into equal groups or finding largest tile size. LCM is the smallest common multiple — useful when finding when events meet again or smallest number divisible by given set. A core identity links them both and is tested directly.
- HCF x LCM = Product of two numbers (only valid for exactly TWO numbers).
- HCF of fractions = HCF of numerators divided by LCM of denominators.
- LCM of fractions = LCM of numerators divided by HCF of denominators.
- Number of factors of N = (a+1)(b+1)(c+1)... if N = p^a x q^b x r^c.
- Sum of all factors formula: [(p^(a+1)-1)/(p-1)] x [(q^(b+1)-1)/(q-1)] x ...
Key formulas
HCF x LCM Identity
HCF(a,b) x LCM(a,b) = a x b
When: When one of HCF, LCM, or one number is unknown — very common in RRB NTPC.
Count of Factors
If N = p^a * q^b * r^c, then total factors = (a+1)(b+1)(c+1)
When: Questions asking how many factors does 360 have, etc.
Worked examples
HCF of two numbers is 12, LCM is 180, one number is 36. Find other. Other = (12 x 180) / 36 = 60.
How many factors does 72 have? 72 = 2^3 x 3^2. Factors = (3+1)(2+1) = 12.
⚠ Common mistakes to avoid
- Treating 1 as a prime number — 1 is neither prime nor composite. This is a direct MCQ trap.
- Using HCF x LCM = product formula for three or more numbers — it only works for exactly two numbers.
- Forgetting that when remainder is 0 in the unit digit cycle, you must pick the LAST element of the cycle, not the first.
- Confusing divisibility by 4 (last 2 digits) with divisibility by 8 (last 3 digits) — swapping these wastes easy marks.
- In fraction HCF/LCM, swapping the formula — students often put HCF of denominators in HCF of fractions formula when it should be LCM of denominators.
🧠 Memory aids
- MNEMONIC for unit digit cycles — 2,3,7,8 have cycle 4. 4,9 have cycle 2. 0,1,5,6 are constant. Think: BIG FOUR (2,3,7,8), SMALL TWO (4,9), STUBBORN FOUR (0,1,5,6).
- For divisibility by 11 — think ODD minus EVEN. Positions, not digits themselves. Count from the LEFT.
- HCF x LCM = product — think of it as a SEESAW. Two numbers on one side, their HCF and LCM on the other. Balance always holds (only for two numbers).
- For remainder shortfall problems — if N is always k short of being perfectly divisible, remember: LCM minus k. Think: ALMOST THERE, MINUS THE GAP.
🎯 RRB NTPC exam tips
- RRB NTPC Phase 1 typically has 2-3 questions directly from unit digit, divisibility rules, and factor counting. These are the fastest-to-solve questions — attempt them first to save time for tougher Quant questions.
- Remainder questions tend to be 1-2 per paper and are slightly harder. Practise the LCM-based shortcut pattern with same-remainder and shortfall types — these two patterns cover 80% of such PYQs.
- Type of numbers (prime, composite, co-prime) appears in GA-crossover style questions too — for example, listing primes between 80 and 100. Know them: 83, 89, 97.
- HCF-LCM identity questions always give 3 out of 4 values and ask for the 4th. Set up the formula immediately and cross-multiply — these should take under 30 seconds.
- Avoid spending more than 60 seconds on any Number System question in the exam. If a remainder question feels long, use options: substitute each option into the condition and check — backsolving is faster than forward calculation for MCQs.