Edit Chapter

Title

Slug (optional)

Summary (optional)

Render Mode

Markdown

HTML (WYSIWYG)

Markdown Content

Supports tables, fenced code, footnotes, checklists, etc.

HTML Content

<div class="d-flex align-items-center gap-2 mb-2">
    <span class="badge text-bg-primary">Quantitative Aptitude</span>
    <span class="badge text-bg-light border">Number System</span>
  </div>
  <p class="text-muted">Basics of numbers, place value, divisibility, HCF/LCM, remainders, and base conversions.</p>
  <hr class="mt-3">

<div class="row g-3">
    <div class="col-12 col-lg-8">
      <div class="card shadow-sm">
        <div class="card-body">
          <h4 class="card-title mb-2">What is the Number System?</h4>
          <p class="mb-2">
            A <strong>number system</strong> is a way to represent and work with quantities. Competitive exams
            test definitions, shortcuts, and properties—especially divisibility, remainders (mod), HCF/LCM,
            and place value tricks.
          </p>
          <div class="alert alert-info small mb-0">
            <strong>Quick idea:</strong> Classify first, then apply rules. Most problems simplify with
            <em>type</em> (prime/composite, even/odd), <em>factorization</em>, or <em>mod</em> arithmetic.
          </div>
        </div>
      </div>
    </div>

<div class="col-12 col-lg-4">
      <div class="card border-0 bg-light h-100">
        <div class="card-body">
          <h6 class="text-uppercase text-muted">Key sets &amp; terms</h6>
          <ul class="mb-2">
            <li><strong>ℕ</strong>: Natural (1,2,3,…)</li>
            <li><strong>𝕎</strong>: Whole (0,1,2,…)</li>
            <li><strong>ℤ</strong>: Integers (…−2,−1,0,1,…)</li>
            <li><strong>ℚ</strong>: Rationals (p/q, q≠0)</li>
            <li><strong>ℝ</strong>: Reals (rationals + irrationals)</li>
            <li><strong>Prime</strong>: &gt;1 with exactly two factors</li>
          </ul>
          <div class="small">
            <span class="badge rounded-pill text-bg-secondary me-1 mb-1">even/odd</span>
            <span class="badge rounded-pill text-bg-secondary me-1 mb-1">prime</span>
            <span class="badge rounded-pill text-bg-secondary me-1 mb-1">composite</span>
            <span class="badge rounded-pill text-bg-secondary me-1 mb-1">coprime</span>
          </div>
        </div>
      </div>
    </div>
  </div>

<div class="row g-3 mt-3">
    <div class="col-12 col-xl-6">
      <div class="card shadow-sm h-100">
        <div class="card-body">
          <h4 class="card-title mb-2">Place Value &amp; Representation</h4>
          <p class="mb-2">In base-10, each digit’s value depends on its position (×10, ×100, …).</p>
          <div class="table-responsive">
            <table class="table table-sm align-middle">
              <thead class="table-light">
                <tr>
                  <th>Number</th>
                  <th>Expanded Form</th>
                  <th>Notes</th>
                </tr>
              </thead>
              <tbody>
                <tr>
                  <td><strong>5,708</strong></td>
                  <td>5×10³ + 7×10² + 0×10¹ + 8×10⁰</td>
                  <td>Zero can hold a place but adds no value.</td>
                </tr>
                <tr>
                  <td><strong>203.45</strong></td>
                  <td>2×10² + 0×10¹ + 3×10⁰ + 4×10⁻¹ + 5×10⁻²</td>
                  <td>Right of decimal → negative powers of 10.</td>
                </tr>
                <tr>
                  <td><strong>n = 10a + b</strong></td>
                  <td>a = tens digit, b = units digit</td>
                  <td>Useful in mod 9 / digit-sum problems.</td>
                </tr>
              </tbody>
            </table>
          </div>

<div class="alert alert-warning small mt-2 mb-0">
            <strong>Tip:</strong> Trailing zeros in <em>N!</em> = number of times 10 divides it.
            Count 5s: <code>⌊N/5⌋ + ⌊N/25⌋ + ⌊N/125⌋ + …</code>
          </div>
        </div>
      </div>
    </div>

<div class="col-12 col-xl-6">
      <div class="card shadow-sm h-100">
        <div class="card-body">
          <h4 class="card-title mb-2">Common Divisibility Rules</h4>
          <div class="table-responsive">
            <table class="table table-sm align-middle">
              <thead class="table-light">
                <tr>
                  <th>k</th>
                  <th>Rule (base-10)</th>
                  <th>Example</th>
                </tr>
              </thead>
              <tbody>
                <tr>
                  <td><strong>2</strong></td>
                  <td>Last digit even</td>
                  <td>748 <span class="text-success">✓</span></td>
                </tr>
                <tr>
                  <td><strong>3</strong></td>
                  <td>Sum of digits divisible by 3</td>
                  <td>7+4+9=20 → no</td>
                </tr>
                <tr>
                  <td><strong>4</strong></td>
                  <td>Last two digits divisible by 4</td>
                  <td>3<strong>12</strong> → yes</td>
                </tr>
                <tr>
                  <td><strong>5</strong></td>
                  <td>Ends with 0 or 5</td>
                  <td>985 <span class="text-success">✓</span></td>
                </tr>
                <tr>
                  <td><strong>8</strong></td>
                  <td>Last three digits divisible by 8</td>
                  <td>1<strong>024</strong> → yes</td>
                </tr>
                <tr>
                  <td><strong>9</strong></td>
                  <td>Digit sum divisible by 9</td>
                  <td>3+6+9=18 → yes</td>
                </tr>
                <tr>
                  <td><strong>11</strong></td>
                  <td>Alt. sum of digits divisible by 11</td>
                  <td>(1−2+1)=0 → yes for 121</td>
                </tr>
              </tbody>
            </table>
          </div>
          <p class="small text-muted mb-0">Use prime factorization for combined checks (e.g., 12 = 3×4).</p>
        </div>
      </div>
    </div>
  </div>

<div class="row g-3 mt-3">
    <div class="col-12 col-lg-7">
      <div class="card shadow-sm h-100">
        <div class="card-body">
          <h4 class="card-title mb-2">HCF/LCM Essentials</h4>
          <ul class="mb-2">
            <li><strong>HCF (GCD)</strong>: greatest common divisor</li>
            <li><strong>LCM</strong>: least common multiple</li>
            <li><strong>Product relation</strong>: <code>a×b = HCF(a,b) × LCM(a,b)</code> (for positive integers)</li>
          </ul>

<div class="row g-2">
            <div class="col-md-6">
              <div class="border rounded p-2 h-100">
                <div class="fw-semibold mb-1">Example</div>
                <div class="small">Find HCF and LCM of 72 and 108.</div>
                <ul class="small mb-0">
                  <li>72 = 2³×3²</li>
                  <li>108 = 2²×3³</li>
                  <li>HCF = 2²×3² = 36</li>
                  <li>LCM = 2³×3³ = 216</li>
                </ul>
              </div>
            </div>
            <div class="col-md-6">
              <pre class="mb-0 p-2 border rounded small"><code>// Euclid GCD (pseudo-code)
gcd(a, b):
  while b ≠ 0:
    (a, b) = (b, a mod b)
  return a</code></pre>
            </div>
          </div>

<div class="alert alert-success small mt-2 mb-0">
            <strong>Coprime trick:</strong> If <em>a</em> and <em>b</em> are coprime, then
            <code>LCM(a,b)=a×b</code> and <code>HCF(a,b)=1</code>.
          </div>
        </div>
      </div>
    </div>

<div class="col-12 col-lg-5">
      <div class="card shadow-sm h-100">
        <div class="card-body">
          <h4 class="card-title mb-2">Remainders &amp; Congruence (mod)</h4>
          <ul class="small mb-2">
            <li><strong>a ≡ b (mod m)</strong> means m divides (a−b).</li>
            <li>Work with remainders: reduce early to keep numbers small.</li>
            <li>(a+b) mod m = ((a mod m)+(b mod m)) mod m</li>
            <li>(a·b) mod m = ((a mod m)·(b mod m)) mod m</li>
            <li>a&lt;sup&gt;k&lt;/sup&gt; mod m often uses patterns/cycles.</li>
          </ul>
          <div class="border rounded p-2 small">
            <div class="fw-semibold">Example</div>
            <div>Find last digit of 7&lt;sup&gt;35&lt;/sup&gt;.</div>
            <div>Pattern of last digit (mod 10): 7,9,3,1 → cycle length 4.</div>
            <div>35 mod 4 = 3 ⇒ last digit is the 3rd in cycle = <strong>3</strong>.</div>
          </div>
        </div>
      </div>
    </div>
  </div>

<div class="row g-3 mt-3">
    <div class="col-12 col-lg-6">
      <div class="card h-100">
        <div class="card-body">
          <h5 class="card-title">Base-n Conversion (Quick)</h5>
          <div class="row row-cols-1 row-cols-md-2 g-2">
            <div class="col">
              <div class="p-2 border rounded">
                <div class="fw-semibold">Binary → Decimal</div>
                <div class="small">e.g., (1011)&lt;sub&gt;2&lt;/sub&gt; = 1·8 + 0·4 + 1·2 + 1·1 = <em>11</em></div>
              </div>
            </div>
            <div class="col">
              <div class="p-2 border rounded">
                <div class="fw-semibold">Decimal → Binary</div>
                <div class="small">Repeated ÷2, collect remainders upward.</div>
              </div>
            </div>
            <div class="col">
              <div class="p-2 border rounded">
                <div class="fw-semibold">Octal/Hex</div>
                <div class="small">Group binary (3 bits for octal, 4 for hex) to convert quickly.</div>
              </div>
            </div>
            <div class="col">
              <div class="p-2 border rounded">
                <div class="fw-semibold">Place Form</div>
                <div class="small">(d&lt;sub&gt;k&lt;/sub&gt;…d&lt;sub&gt;1&lt;/sub&gt;d&lt;sub&gt;0&lt;/sub&gt;)&lt;sub&gt;b&lt;/sub&gt; = Σ d&lt;sub&gt;i&lt;/sub&gt;·b&lt;sup&gt;i&lt;/sup&gt;</div>
              </div>
            </div>
          </div>
          <p class="small text-muted mt-2 mb-0">Check digits: each d&lt;sub&gt;i&lt;/sub&gt; must be &lt; base.</p>
        </div>
      </div>
    </div>

<div class="col-12 col-lg-6">
      <figure class="border rounded p-3 bg-light h-100 d-flex flex-column">
        <img src="https://via.placeholder.com/640x360?text=Number+System" alt="Number System concept" class="img-fluid rounded" loading="lazy" width="640" height="360">
        <figcaption class="small text-muted mt-2">
          Visual placeholder. Replace with a chapter diagram if needed.
        </figcaption>
      </figure>
    </div>
  </div>

<div class="row g-3 mt-4">
    <div class="col-12">
      <h4 class="fw-bold">Practice</h4>
    </div>

<div class="col-12 col-lg-6">
      <div class="card shadow-sm h-100">
        <div class="card-body">
          <h5 class="card-title">A) Multiple Choice</h5>
          <ol class="mb-0">
            <li class="mb-2">
              The units digit of 3&lt;sup&gt;57&lt;/sup&gt; is:
              <div class="mt-1">
                <span class="badge rounded-pill text-bg-light border me-1">1</span>
                <span class="badge rounded-pill text-bg-light border me-1">3</span>
                <span class="badge rounded-pill text-bg-light border me-1">7</span>
                <span class="badge rounded-pill text-bg-light border me-1">9</span>
              </div>
            </li>
            <li class="mb-2">
              HCF(84, 126) equals:
              <div class="mt-1">
                <span class="badge rounded-pill text-bg-light border me-1">14</span>
                <span class="badge rounded-pill text-bg-light border me-1">21</span>
                <span class="badge rounded-pill text-bg-light border me-1">42</span>
                <span class="badge rounded-pill text-bg-light border me-1">63</span>
              </div>
            </li>
            <li class="mb-2">
              The number divisible by 11 is:
              <div class="mt-1">
                <span class="badge rounded-pill text-bg-light border me-1">1,232</span>
                <span class="badge rounded-pill text-bg-light border me-1">1,254</span>
                <span class="badge rounded-pill text-bg-light border me-1">1,331</span>
                <span class="badge rounded-pill text-bg-light border me-1">1,421</span>
              </div>
            </li>
          </ol>
        </div>
      </div>
    </div>

<div class="col-12 col-lg-6">
      <div class="card shadow-sm h-100">
        <div class="card-body">
          <h5 class="card-title">B) Fill in the Blanks</h5>
          <ol class="mb-0">
            <li class="mb-2">Trailing zeros in 100! = _______. <em>(use powers of 5)</em></li>
            <li class="mb-2">If a ≡ 7 (mod 9) and b ≡ 5 (mod 9), then (a+b) ≡ _______ (mod 9).</li>
            <li class="mb-2">LCM(18, 24) = _______. (use prime factors)</li>
            <li class="mb-2">(10101)&lt;sub&gt;2&lt;/sub&gt; in decimal = _______. </li>
          </ol>
        </div>
      </div>
    </div>

<div class="col-12">
      <div class="card">
        <div class="card-body">
          <h5 class="card-title mb-2">C) Tiny code sample (formatting test)</h5>
          <pre class="mb-0"><code>// Fast digit cycle for last digit (mod 10)
const lastDigit = (a, n) =&gt; {
  const cycle = [a%10];
  while (true) {
    const next = (cycle.at(-1) * (a%10)) % 10;
    if (next === cycle[0]) break;
    cycle.push(next);
  }
  return cycle[(n-1) % cycle.length];
};</code></pre>
        </div>
      </div>
    </div>
  </div>

<div class="row g-3 mt-4">
    <div class="col-12">
      <div class="card border-0 bg-light">
        <div class="card-body">
          <h5 class="card-title">Short Reading</h5>
          <p class="mb-2">
            A factory produces items in batches of <strong>18</strong> and packs them into boxes of
            <strong>24</strong>. To combine batches with no leftovers, they must produce a count that is a
            multiple of both—i.e., the <strong>LCM</strong>. For quick checks, use prime factorization:
            18 = 2×3² and 24 = 2³×3 ⇒ LCM = 2³×3² = 72.
          </p>
          <p class="small text-muted mb-0">Task: If each item weighs 250 g, how many kilograms are in the minimum combined shipment?</p>
        </div>
      </div>
    </div>

<div class="col-12 col-lg-8">
      <details class="border rounded">
        <summary class="p-3 fw-semibold">Show Suggested Answers</summary>
        <div class="p-3">
          <h6 class="fw-semibold">MCQ</h6>
          <ol>
            <li><strong>7</strong> (cycle of 3: 3,9,7,1; 57 mod 4 = 1 → 3; wait, for 3 the cycle is 3,9,7,1 → 57 mod 4 = 1 → <em>3</em>)</li>
            <li><strong>42</strong> (84=2²·3·7; 126=2·3²·7 → HCF=2·3·7)</li>
            <li><strong>1,331</strong> (1−3+3−1=0 → divisible by 11)</li>
          </ol>
          <h6 class="fw-semibold">Fill in the Blanks</h6>
          <ol>
            <li><strong>24</strong> (⌊100/5⌋=20, ⌊100/25⌋=4; total 24)</li>
            <li><strong>3</strong> (7+5=12 ≡ 3 mod 9)</li>
            <li><strong>72</strong> (18=2·3², 24=2³·3 → LCM=2³·3²)</li>
            <li><strong>21</strong> (16+0+4+0+1)</li>
          </ol>
          <h6 class="fw-semibold">Reading</h6>
          <p class="mb-0 small">
            Minimum shipment = LCM count = 72 items; weight = 72×0.25 kg = <em>18 kg</em>.
          </p>
        </div>
      </details>
    </div>

<div class="col-12 col-lg-4">
      <div class="card border-0 bg-light h-100">
        <div class="card-body">
          <h6 class="text-uppercase text-muted">Exam tips</h6>
          <ul class="small mb-0">
            <li>Reduce numbers early with mod.</li>
            <li>Use prime powers for HCF/LCM quickly.</li>
            <li>Memorize small digit cycles (2,3,4,7,8,9).</li>
            <li>Check divisibility before long division.</li>
          </ul>
        </div>
      </div>
    </div>
  </div>

</div>

Rich formatting, tables, media embeds. (Server sanitizes HTML with Bleach.)

Topic

Sort Order

Active

Cancel

Learning Modules

Main Menu

Learning Modules

Settings

Edit Chapter