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Number System (Advanced)

The ₹1 Crore Question Hidden in a Remainder

A candidate stares at this question: *"What is the remainder when 2^100 is divided by 7?"* He starts multiplying 2 × 2 × 2... and runs out of time. Another candidate solves it in 15 seconds — because she knows one simple pattern.

That is the entire secret of the Advanced Number System. It is not about harder calculation. It is about seeing patterns that others miss. In exams like SSC CGL, IBPS PO, and RRB NTPC, this chapter appears regularly — and most students lose those marks not because the maths is hard, but because nobody taught them the *shortcuts behind the shortcuts*.

Today, that changes.

Learning Goals

By the end of this chapter, you will be able to:

1. Classify numbers instantly (prime, composite, co-prime, perfect).
2. Use divisibility rules — including the tricky ones for 7, 11, and 13.
3. Find remainders of huge powers using cyclicity and the Remainder Theorem.
4. Count factors, sum factors, and find unit digits in seconds.
5. Handle HCF–LCM word problems and base conversions with confidence.

1. The Family Tree of Numbers

Every number you'll ever meet in an exam belongs to a family:

• Natural numbers (N): 1, 2, 3, ... — the counting numbers.
• Whole numbers (W): 0, 1, 2, 3, ... — natural numbers plus zero.
• Integers (Z): ..., −2, −1, 0, 1, 2, ...
• Rational numbers (Q): numbers expressible as p/q, where q ≠ 0. Their decimals either *terminate* (0.25) or *repeat* (0.333...).
• Irrational numbers: cannot be written as p/q — √2, √3, π. Their decimals never end and never repeat.
• Real numbers (R): rational + irrational — the complete number line.

Exam trap: Is 0.999... (repeating) rational? Yes! It equals exactly 1. Any repeating decimal is rational.

🧠 Active Recall: Close the book. Is √9 irrational? (No — √9 = 3, a rational number. The root must be *non-terminating* to be irrational.) Is every integer a whole number? (No — negative integers are not whole numbers.)

2. Prime, Composite, and Co-prime — The VIP Club

• Prime number: has exactly two factors — 1 and itself. Examples: 2, 3, 5, 7, 11, 13...
• Composite number: more than two factors. Examples: 4, 6, 8, 9...
• 1 is neither prime nor composite. (Examiners love this trap!)
• 2 is the only even prime. Every other even number is divisible by 2.

Co-prime numbers are pairs whose HCF is 1 — they share no common factor except 1. Example: (8, 15) are co-prime even though neither is prime. Co-prime ≠ prime!

How many primes below 100? Exactly 25. Memorize this — it appears directly in exams.

🧠 Active Recall: Without looking — is 91 prime? (No! 91 = 7 × 13. This is the most common "fake prime" in exams. Others: 51 = 3 × 17, 57 = 3 × 19, 87 = 3 × 29.)

3. Divisibility Rules — Including the Ones Nobody Teaches

You know 2, 3, 5, 10. Here is the complete arsenal:

Example (7): Is 203 divisible by 7? Take 20 − (2×3) = 20 − 6 = 14. Yes, 14 is divisible by 7. ✓

Example (11): Is 918082 divisible by 11? Odd places: 9+8+8 = 25. Even places: 1+0+2 = 3. Difference = 22, a multiple of 11. ✓

🧠 Active Recall: Test 1001 for divisibility by 7, 11, and 13. (Surprise: 1001 = 7 × 11 × 13 — divisible by all three! This fact alone solves many exam questions.)

4. Factors — Counting Them Like a Pro

This is where toppers separate from the crowd. For any number, write its prime factorization:

N = a^p × b^q × c^r (where a, b, c are primes)

Then:

• Number of factors = (p+1)(q+1)(r+1)
• Sum of factors = (a⁰+a¹+...+a^p)(b⁰+...+b^q)(c⁰+...+c^r)
• Number of even factors = p × (q+1)(r+1) — if a = 2
• Number of odd factors = (q+1)(r+1) — just ignore the power of 2
• Product of factors = N^(number of factors / 2)

Example: N = 360 = 2³ × 3² × 5¹

• Total factors = (3+1)(2+1)(1+1) = 4×3×2 = 24
• Odd factors = (2+1)(1+1) = 6
• Even factors = 24 − 6 = 18

Perfect squares always have an odd number of factors (one factor pairs with itself). This is a favourite statement-based question.

🧠 Active Recall: How many factors does 72 have? (72 = 2³ × 3² → (3+1)(2+1) = 12.) Does 144 have an odd or even count of factors? (Odd — it's a perfect square, 12².)

5. Unit Digit and Cyclicity — The 15-Second Superpower

Remember the opening story? Here is the trick. The unit digits of powers repeat in cycles:

Method: Divide the exponent by the cycle length. Use the remainder to pick the position. (If remainder = 0, take the *last* position of the cycle.)

Example: Unit digit of 7^103? Cycle length 4. 103 ÷ 4 → remainder 3. Third position in (7, 9, 3, 1) = 3.

🧠 Active Recall: Unit digit of 2^100? (100 ÷ 4 → remainder 0 → last position of 2,4,8,6 → 6.)

6. Remainders — The Heart of Advanced Number System

Basic rule: Remainders behave like the numbers themselves under +, −, ×.

If a ÷ n leaves remainder r₁ and b ÷ n leaves remainder r₂, then:

• (a + b) ÷ n leaves remainder of (r₁ + r₂) ÷ n
• (a × b) ÷ n leaves remainder of (r₁ × r₂) ÷ n

Negative remainders: If remainder is −1, treat it as (divisor − 1). This makes huge problems tiny.

Example: Remainder when 8^100 is divided by 7?
8 ≡ 1 (mod 7), so 8^100 ≡ 1^100 = 1. Done in 5 seconds.

Example: Remainder when 2^100 ÷ 7? Powers of 2 mod 7: 2, 4, 1, 2, 4, 1... cycle of 3. 100 ÷ 3 → remainder 1 → answer 2. (This was the opening story's question!)

Fermat's Little Theorem: If p is prime and a is not divisible by p, then a^(p−1) ÷ p leaves remainder 1. Example: 2^6 ÷ 7 → remainder 1.

🧠 Active Recall: Remainder when 9^50 ÷ 8? (9 ≡ 1 mod 8 → answer 1.) Remainder when 5^3 ÷ 7? (125 ÷ 7 = 17×7 + 6 → 6.)

7. HCF and LCM — Beyond the Basics

• HCF × LCM = Product of the two numbers (works only for *two* numbers — a classic trap!).
• HCF of fractions = HCF of numerators / LCM of denominators.
• LCM of fractions = LCM of numerators / HCF of denominators.
• If two numbers are in ratio a : b (co-prime) with HCF = h, the numbers are ah and bh, and LCM = abh.

Word-problem decoder:

• "Greatest number that divides..." → HCF
• "Smallest number divisible by..." → LCM
• "Bells toll together again" → LCM
• "Maximum size of equal pieces" → HCF

🧠 Active Recall: HCF × LCM for three numbers — does the product formula work? (No! Only for two numbers. Examiners test exactly this.)

8. Base Systems — The Silent Scorer

We live in base 10, but exams visit other bases:

• Decimal → other base: Divide repeatedly by the base, collect remainders bottom-to-top.
• Other base → decimal: Multiply each digit by base^position and add.

Example: (1011)₂ = 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = (11)₁₀

🧠 Active Recall: Convert 25 to binary. (25 = 16+8+1 → 11001.)

🎭 The Mnemonic You'll Never Forget

"Lazy Elephants Jump Slowly" — for the unit-digit cycle of 2: 2, 4, 8, 6 — imagine 2 Lazy elephants, then 4, then 8, then 6, jumping slowly in a loop forever. Weird? Yes. Unforgettable? Also yes.

And for the fake primes: "91 is a LIAR" — it looks prime, smells prime, but 91 = 7 × 13. Whenever a number *feels* prime in an exam, interrogate it!

📌 Summary Table

🔁 Review Reminder

Review this chapter after 2 days. Solve just 5 questions: one on factors, one on unit digits, one on remainders, one on HCF-LCM, one on divisibility. If you can do those without peeking, the patterns have moved into permanent memory — and those exam marks are yours.