Classification of Numbers & Divisibility Rules

Number system questions are the silent backbone of every RPF Sub-Inspector mathematics paper — primes, co-primes, even/odd, rationals — they all turn into one-line answers if you have the family tree clearly drawn in your head.

Definition: Natural numbers (N) are the counting numbers: 1, 2, 3, 4, … . They start from 1 and never include 0 in the Indian school convention used by RPF/SSC/Railway exams.

Definition: Whole numbers (W) are the natural numbers together with 0: 0, 1, 2, 3, … . They are exactly N with zero added.

Definition: Integers (Z) include all whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, … . Z comes from the German Zahl meaning "number".

Definition: Rational numbers (Q) are numbers that can be written as p/q where p and q are integers and q ≠ 0. Every integer is a rational number (e.g., 5 = 5/1). Their decimal expansions either terminate (e.g., 0.75) or recur (e.g., 0.333…).

Definition: Irrational numbers cannot be written as p/q. Their decimal expansions are non-terminating and non-recurring. Examples: √2 ≈ 1.41421356…, π ≈ 3.14159265…, e ≈ 2.71828182….

Definition: Real numbers (R) are the union of all rationals and irrationals; every point on the number line is a real number.

Definition: A prime number is a natural number greater than 1 that has exactly two distinct positive factors — 1 and itself. The smallest prime is 2, and it is the only even prime.

Definition: A composite number is a natural number with more than two distinct positive factors (e.g., 4, 6, 8, 9, 10).

Definition: Co-prime (relatively prime) numbers are two numbers whose HCF (greatest common factor) is 1. They need not themselves be prime — for example, 8 and 15 are co-prime even though both are composite.

The number family tree, drawn once

Think of the number families as a series of larger and larger circles. The innermost circle is N (1, 2, 3…). Wrap a slightly bigger circle around it and add 0 — now you have W. Wrap another circle and add the negatives — that gives Z. Wrap one more and add all p/q forms — that gives Q. Outside Q live the irrationals. Q together with the irrationals is the universe R.

This nesting is testable directly: "Is −3 a whole number?" No, because W stops at 0 and goes upward; −3 lives in Z but not W. "Is 7 a rational number?" Yes, because 7 = 7/1. "Is √4 irrational?" No, because √4 = 2, which is a perfectly good integer.

Primes, composites and the one in the middle

Prime numbers are the building blocks of every natural number (Fundamental Theorem of Arithmetic): every integer greater than 1 factors uniquely into primes. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. There are exactly 25 primes below 100, a constant that RPF, SSC and Railway exams have asked verbatim more than once. Memorise the number, and ideally memorise the primes themselves up to 50.

The number 1 is the odd one out: it is neither prime nor composite. It has only one factor (itself), so it fails the "exactly two factors" rule for prime and the "more than two factors" rule for composite.

The number 2 is also special — it is the only even prime. Every other even number is divisible by 2 (apart from itself and 1) and is therefore composite.

Co-prime, not "both prime"

Co-prime asks only about the HCF. The pair (8, 15) is co-prime because HCF(8, 15) = 1, even though 8 and 15 are both composite. Conversely, two primes are always co-prime, but the converse is not required.

Why it matters: RPF and SSC question-setters love trap options like "co-prime means both numbers are prime". A single second of confusion here can cost you a mark.

Real-world example: A railway ticket counter shows ticket numbers that increment by either 7 or 9 on adjacent counters. Since HCF(7, 9) = 1, the two counters' ticket numbers are co-prime — and they will only repeat together after 7 × 9 = 63 tickets. The LCM = HCF × … identity behind this is the same one that makes co-prime so practically useful.

Common misconception: "Every even number is composite." Wrong — 2 is even and prime. Also wrong: "Every prime is odd." Wrong for the same reason.

Common misconception: "Recurring decimals like 0.333… are irrational." Wrong — they are rational, equal to 1/3.

Question: Classify each of the following: −5, 0, 1, 2, 4, √7, 22/7, 0.272727…

Solution:
Step 1: −5 — integer, rational, real. Not natural, not whole, not prime, not composite.
Step 2: 0 — whole, integer, rational, real. Not natural, not prime, not composite.
Step 3: 1 — natural, whole, integer, rational, real. Neither prime nor composite.
Step 4: 2 — natural, whole, integer, rational, real, and the only even prime.
Step 5: 4 — natural, whole, integer, rational, real, composite (factors 1, 2, 4).
Step 6: √7 — real, irrational (cannot be written as p/q).
Step 7: 22/7 — rational (it is a fraction). Note that 22/7 is not π; it is only an approximation.
Step 8: 0.272727… — rational, since the bar makes it 27/99 = 3/11.
Conclusion: Tag every number with as many family labels as fit; the smallest set is most specific.

By 2: last digit even. By 3: digit-sum divisible by 3. By 4: last two digits divisible by 4. By 5: ends in 0 or 5. By 6: divisible by both 2 and 3. By 8: last three digits divisible by 8. By 9: digit-sum divisible by 9. By 10: ends in 0. By 11: (sum of odd-place digits) - (sum of even-place digits) = 0 or multiple of 11. By 25: last two digits 00,25,50,75. Shortcut for 7: double the last digit, subtract from the rest; if result divisible by 7, the number is too (e.g., 203: 20 - 6 = 14, divisible by 7).

Check if 4,832,718 is divisible by 11. Mark places from the right: digits 8(1),1(2),7(3),2(4),3(5),8(6),4(7). Odd-place (1,3,5,7) digits: 8+7+3+4 = 22. Even-place (2,4,6) digits: 1+2+8 = 11. Difference = 22 - 11 = 11, which is a multiple of 11, so the number IS divisible by 11. This alternating-sum method is far faster than long division and is the most-tested divisibility rule in RPF SI papers, so practice tagging odd/even positions from the rightmost digit.

Cover the answer, recall, then check. 12 cards on number types and divisibility rules for RPF SI.

Q1. State the divisibility rule for 8.
A1. The number formed by the last 3 digits is divisible by 8. E.g. 17,416 → 416 ÷ 8 = 52, so divisible.

Q2. Divisibility rule for 11?
A2. (Sum of digits at odd places) − (sum at even places) is 0 or a multiple of 11. E.g. 4,832: (2+8) − (3+4) = 3 → not divisible.

Q3. Fast rule for 7?
A3. Double the last digit and subtract from the rest; repeat. 672 → 67 − 2×2 = 63 = 7×9 → divisible.

Q4. How many prime numbers are there below 20, and list them.
A4. 8 primes: 2, 3, 5, 7, 11, 13, 17, 19.

Q5. Is 1 prime, composite, or neither?
A5. Neither — a prime has exactly two factors; 1 has only one factor.

Q6. Divisibility rule for 12?
A6. Divisible by BOTH 3 (digit sum) and 4 (last two digits). Must satisfy both.

Q7. Rule for 6?
A7. Divisible by 2 (even) and by 3 (digit sum divisible by 3) together.

Q8. Sum of first n natural numbers?
A8. n(n+1)/2. Sum 1..100 = 100×101/2 = 5050.

Q9. What is the unit digit of 7^84?
A9. Cycle of 7 is 7,9,3,1 (length 4). 84 ÷ 4 leaves remainder 0 → 4th in cycle = 1.

Q10. Distinguish rational vs irrational.
A10. Rational = p/q form (q≠0), terminating/repeating decimal. Irrational (√2, π) is non-terminating non-repeating.

Q11. Divisibility rule for 13?
A11. Multiply last digit by 4, add to the remaining number; repeat. 195 → 19 + 5×4 = 39 = 13×3 → divisible.

Q12. Number of factors of 72?
A12. 72 = 2³×3² → factors = (3+1)(2+1) = 12.

Number System is the silent backbone of RPF SI Mathematics. Directly, 1–2 questions test divisibility, unit digits, or number of factors; indirectly, every Simplification, HCF/LCM and remainder problem leans on these rules. Since SI has a higher cutoff than Constable, these are marks you cannot afford to leak — they are quick, mechanical, and calculator-free.

Why it matters

A divisibility trick replaces a full long division; a unit-digit cycle answers "last digit of 3^47" in seconds; factor-counting cracks "how many divisors" instantly.

Classification at a glance

Type	Definition	Example
Natural	Counting numbers	1, 2, 3, …
Whole	Naturals + 0	0, 1, 2, …
Prime	Exactly two factors	2, 3, 5, 7
Composite	More than two factors	4, 6, 9, 12
Rational	p/q form, q≠0	3/4, 0.25, −7
Irrational	Non-terminating, non-repeating	√2, π

Core divisibility rules

• 2: last digit even. 4: last two digits divisible by 4. 8: last three digits divisible by 8.
• 3 / 9: digit sum divisible by 3 / 9.
• 5: ends in 0 or 5. 6: divisible by 2 and 3. 12: by 3 and 4.
• 11: (odd-place sum − even-place sum) is 0 or multiple of 11.
• 7: double last digit, subtract from rest, repeat.

Unit-digit cycles

2→(2,4,8,6), 3→(3,9,7,1), 7→(7,9,3,1), 8→(8,4,2,6): all length 4. Divide the power by 4 and read the remainder position.

Number of factors

If N = a^p · b^q · c^r, factors = (p+1)(q+1)(r+1). E.g. 72 = 2³·3² → 12 factors.

Exam Tricks & Tips

• 🎯 For "divisible by 12/15/18", split into coprime parts (3&4, 3&5, 2&9) and test each — faster than trial division.
• 🎯 Memorise unit-digit cycles; remainder 0 means the LAST term of the cycle, not the first.
• 🎯 Sum 1..n = n(n+1)/2; sum of first n odd numbers = n².
• 🎯 A perfect square never ends in 2, 3, 7 or 8 — reject such options.
• 🎯 To count factors quickly, always prime-factorise first; add 1 to each exponent and multiply.
• ❌ Common mistake: treating 1 as prime, or forgetting 2 is the only even prime — both wreck prime-counting.

Expected exam pattern

1–2 direct questions: identify the divisor, find the unit digit of a large power, count factors, or classify a number. Distractors exploit the "1 is prime" and "perfect-square unit-digit" traps.

Quick recap

Learn the rules for 2–13, the four length-4 unit-digit cycles, n(n+1)/2, and the (p+1)(q+1)… factor formula. These convert slow arithmetic into instant answers.