1  Number Systems

1.1 Introduction

The number system is the foundation of quantitative aptitude. Many exam questions on arithmetic, divisibility, remainders, and modular arithmetic stem from this topic.
This chapter builds a comprehensive toolkit covering: - Classification of numbers (natural, whole, integers, rationals, irrationals, reals).
- Properties: even–odd, prime–composite, coprimality.
- Divisibility rules, LCM and HCF.
- Remainders and modular arithmetic.
- Factorization and highest powers.
- Base systems, digital sums, cyclicity of remainders.
- Standard exam applications with solved examples and practice sets.

1.2 Classification of Numbers

  1. Natural Numbers (N): \(1,2,3,\dots\)
  2. Whole Numbers (W): \(0,1,2,3,\dots\)
  3. Integers (Z): \(\dots,-2,-1,0,1,2,\dots\)
  4. Rational Numbers (Q): \(\frac{p}{q}\), \(q\neq0\)
  5. Irrational Numbers: Cannot be written as ratio, e.g., \(\sqrt{2},\pi,e\).
  6. Real Numbers (R): Rationals \(\cup\) Irrationals.
  7. Prime Numbers: >1, divisible by 1 and itself.
  8. Composite Numbers: Non-prime, >1.

1.3 Even–Odd Properties

  • Even ± Even = Even
  • Even ± Odd = Odd
  • Odd ± Odd = Even
  • Even × Anything = Even
  • Odd × Odd = Odd

Example: \(37^2-36^2\) = \((37-36)(37+36)=73\) → Odd.

1.4 Divisibility Rules

  • By 2: last digit even.
  • By 3/9: sum of digits divisible.
  • By 4: last two digits divisible by 4.
  • By 5: last digit 0 or 5.
  • By 6: divisible by 2 and 3.
  • By 7: Double last digit, subtract from rest.
  • By 8: last three digits divisible by 8.
  • By 11: Difference of odd–even placed digits divisible by 11.

1.5 LCM and HCF

  • HCF (GCD): Highest common factor.
  • LCM: Lowest common multiple.
  • Relation:
    \[ LCM \times HCF = \text{Product of numbers} \]

Example: LCM(18,24)=72; HCF=6; Product=432 → relation holds.

1.6 Factorization and Highest Power

  • Prime factorization: Express number as product of primes.
  • Highest power of prime \(p\) in \(n!\):
    \[ \sum_{k=1}^\infty \left\lfloor \frac{n}{p^k} \right\rfloor \]

Example: Highest power of 2 in \(100!\) = \(50+25+12+6+3+1=97\).

1.7 Remainders and Modular Arithmetic

If \(N=qd+r\) (Euclidean division):
- \(q=\) quotient, \(r=\) remainder.
- \(N\equiv r \pmod d\).

Rules:
- \((a+b)\bmod m = (a\bmod m + b\bmod m)\bmod m\)
- \((a-b)\bmod m = (a\bmod m - b\bmod m)\bmod m\)
- \((a\cdot b)\bmod m = (a\bmod m \cdot b\bmod m)\bmod m\)

Example: \(23^{45}\bmod 5\).
\(23\equiv3\pmod5\), so need \(3^{45}\bmod5\).
Cycle of \(3^n\bmod5\): 3,4,2,1 → length 4.
\(45\equiv1\pmod4\) → remainder = 3.

1.8 Digital Sum and Casting Out 9

  • Sum of digits of \(N\) ≡ \(N\bmod9\).
  • Useful for checking mistakes.

Example: \(987654 →\) sum=39 → 3+9=12 → 1+2=3 → So \(987654\equiv3\pmod9\).

1.9 Cyclicity of Last Digits

  • Last digit repeats in cycles.
  • Example: powers of 7 → 7,9,3,1 repeat every 4.
  • To find last digit of \(7^{222}\): 222 ÷ 4 → remainder 2 → last digit 9.

1.10 Base Systems

  • Base \(b\): digits 0 to \(b-1\).
  • Value: \((d_n b^n + d_{n-1} b^{n-1}+\dots+d_0)\).

Example: Convert \((1011)_2\) to decimal: \(1\cdot8+0\cdot4+1\cdot2+1=11\).

1.11 Solved Examples

  1. Find remainder when \(2^{100}\) is divided by 7.
    Cyclicity of \(2^n\) mod 7 = 2,4,1. Length=3. \(100\equiv1\) mod 3 → remainder=2.

  2. HCF of 136 and 238.
    Euclidean method: 238-136=102; 136-102=34; 102-3×34=0 → HCF=34.

  3. Find number of zeroes in \(100!\).
    Power of 5: \(100/5+100/25=20+4=24\).

  4. Last two digits of \(7^{222}\).
    Cycle of \(7^n\) mod 100: 7,49,43,1 (length=20). \(222\equiv2\) mod 20 → 49.

  5. If product of two numbers=3780 and HCF=18, find LCM.
    LCM=3780/18=210.

1.12 Common Traps

  • Confusing HCF/LCM formulas.
  • Forgetting remainder must be < divisor.
  • Not reducing base cases in cyclicity.
  • Misinterpreting “divisible by” vs “leaves remainder”.
  • Forgetting 0 is divisible by every integer.

1.13 Practice Set – Level 1

  1. Find HCF and LCM of 42 and 60.
  2. Find remainder of \(5^{100}\) ÷ 13.
  3. Find highest power of 7 in 1000!.
  4. Convert (234) base 5 to decimal.
  5. Find number of divisors of 360.

1.14 Practice Set – Level 2

  1. Find remainder of \(17^{256}\) ÷ 19.
  2. If LCM of two numbers=240 and HCF=12, product of numbers?
  3. Find last digit of \(3^{200}\).
  4. Find number of trailing zeroes in 250!.
  5. Find digital sum of \(987654321\).

1.15 Practice Set – Level 3

  1. Find remainder when \(7^{222}\) ÷ 1000.
  2. If \(a=2^n\) and \(b=5^n\), find HCF and LCM of (a−1,b−1).
  3. In how many ways can 360 be expressed as product of two co-primes?
  4. Find number of divisors of 7560 divisible by 9.
  5. Convert (345) base 7 to decimal and back to base 9.

1.16 Answer Key (Outline)

Level 1: 1) HCF=6, LCM=420. 2) 9. 3) 142. 4) 69. 5) 24 divisors.
Level 2: 6) 1. 7) 2880. 8) 1. 9) 62. 10) 9.
Level 3: 11) 849. 12) HCF=1, LCM=(a−1)(b−1). 13) 16. 14) 60. 15) 180→base 9=(220).

1.17 Summary

  • Master divisibility, factorization, remainders, and digital roots.
  • Always reduce using modular arithmetic.
  • Cyclicity handles powers and last digits.
  • Base conversion skills sharpen place-value understanding.
  • Traps mostly involve misreading the base or remainder definition.