SSC CGL Preparation – Day 1

Quantitative Aptitude: Number System

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🌟 Important Concepts:

1. Natural Numbers: Numbers starting from 1 (1, 2, 3, …).
2. Whole Numbers: Natural numbers + 0 (0, 1, 2, 3, …).
3. Integers: All positive and negative whole numbers including 0 (…, -3, -2, -1, 0, 1, 2, 3, …).
4. Rational Numbers: Numbers that can be expressed as p/q, where p and q are integers and q≠0.
5. Irrational Numbers: Numbers that cannot be expressed as p/q​ $$(e.g., \sqrt{2}​,π).$$
6. Prime Numbers: Numbers greater than 1, divisible only by 1 and itself (2, 3, 5, 7, 11, etc.).
7. Composite Numbers: Numbers with more than two divisors (4, 6, 8, 9, etc.).
8. Even and Odd Numbers: Even (divisible by 2), Odd (not divisible by 2).

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🧠 Key Topics to Study:

1. Divisibility Rules

(Important for fast calculations!)

Number	Divisibility Rule
2	Last digit is 0, 2, 4, 6, 8
3	Sum of digits divisible by 3
4	Last two digits divisible by 4
5	Last digit is 0 or 5
6	Divisible by both 2 and 3
8	Last three digits divisible by 8
9	Sum of digits divisible by 9
10	Last digit is 0
11	Alternating sum/difference of digits divisible by 11

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2. LCM (Least Common Multiple)

• Smallest number divisible by all the given numbers.
• Find prime factors, pick highest powers.

Example:
Find LCM of 12 and 18:
$$12=2^2 \times 3$$
$$18=2 \times 3^2$$
$$LCM = 2^2 \times 3^2 = 36$$

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3. HCF (Highest Common Factor)

• Largest number that divides all given numbers.
• Find prime factors, pick lowest powers.

Example:
Find HCF of 12 and 18:
$$12=2^2 \times 3$$
$$18=2 \times 3^2$$
$$HCF = 2^1 \times 3^1=6$$

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4. Important Properties:

• Product of two numbers = HCF × LCM
• For any prime number p, number of divisors of pⁿ = n+1
• Sum of n natural numbers = $$\frac{n(n+1)}{2}$$
• Sum of squares of first n natural numbers = $$\frac{n(n+1)(2n+1)}{6}​$$

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5. Types of Questions Asked in SSC CGL:

• Find HCF/LCM
• Divisibility Test
• Number of zeros at the end (factorial questions)
• Prime number based questions
• Remainders (special remainder theorems)
• Simplified calculations using divisibility tricks

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📝 Example Questions:

Q1. What is the sum of first 50 natural numbers?
✍️ Solution:
$$\text{Sum} = \frac{n(n+1)}{2} = \frac{50(51)}{2} = 1275$$

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Q2. Find HCF and LCM of 36 and 48.
✍️ Solution:
$$36=2^2 \times 3^2$$
$$48=2^4 \times 3$$

• $$HCF =2^2 \times 3^1 =$$
• $$LCM = 2^4 \times 3^2=144$$

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Q3. Find the number of zeros at the end of 50! (50 factorial).
✍️ Solution:
Zeros are contributed by 5’s in factorization:
$$\text{Zeros} = \left\lfloor \frac{50}{5} \right\rfloor + \left\lfloor \frac{50}{25} \right\rfloor = 10 + 2 = 12$$

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🎯 Quick Tricks:

• LCM Shortcut: If numbers are coprime, LCM = product.
• Finding Last Digit: Focus on unit digits separately.
• Number of divisors:
  If N=$$p^a \times q^b \times r^c$$
  then total divisors = $$(a+1)(b+1)(c+1)$$

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📖 Homework for Practice:

• Find the LCM and HCF of 15, 20, and 30.
• Check if 3780 is divisible by 3, 6, and 9.
• How many zeros are there at the end of 100!?