Number System: Notes और MCQ Practice

आसान भाषा में:

Number System SSC CGL Quant का base है। आसान भाषा में कहें तो, यह numbers को categorise करने और उनकी properties को समझने का खेल है। इसमें Real Numbers, Integers, Prime Numbers, Divisibility Rules और Unit Digit से direct questions पूछे जाते हैं। इस topic पर command मतलब पूरे Quant में speed boost!

Important Formulas & Rules:

• Sum of first 'n' natural numbers = n × (n + 1) ÷ 2
• Sum of first 'n' odd numbers = n²
• Sum of first 'n' even numbers = n × (n + 1)
• Divisibility of 8: Last 3 digits 8 से divisible होने चाहिए।
• Divisibility of 9: Sum of digits 9 से divisible होना चाहिए।

2 Quick Tricks:

1. Unit Digit Cycle of 5, 6, 1, 0: इन numbers की power कुछ भी हो, unit digit हमेशा same (5, 6, 1, 0) ही रहेगी।
2. Unit Digit of 4 and 9: 4 की power odd है तो unit digit 4, even है तो 6. 9 की power odd है तो unit digit 9, even है तो 1.

2 Common Mistakes to Avoid:

1. Confusing 1 as Prime: 1 को Prime Number मान लेना। याद रखें, 1 न तो Prime है और न ही Composite. सबसे छोटा Prime Number 2 है।
2. Divisibility of 6: 6 की divisibility के लिए सिर्फ 2 या 3 check करना। Rule यह है कि number 2 और 3 दोनों से divisible होना चाहिए।

Number System: Concept & Tricks

1. Concept

The Number System is the backbone of Quantitative Aptitude in SSC CGL. It deals with the properties, classification, and operations performed on numbers.

Classification of Numbers:

• Natural Numbers (N): Counting numbers starting from 1. (1, 2, 3, ...)
• Whole Numbers (W): Natural numbers including 0. (0, 1, 2, 3, ...)
• Integers (Z): Positive, negative numbers, and zero. (..., -3, -2, -1, 0, 1, 2, 3, ...)
• Prime Numbers: Numbers greater than 1 that have only two factors: 1 and the number itself. (2, 3, 5, 7, 11, 13, ...) Note: 2 is the only even prime number.
• Composite Numbers: Numbers greater than 1 that have more than two factors. (4, 6, 8, 9, ...) Note: 1 is neither prime nor composite.
• Co-prime Numbers: Two numbers whose Highest Common Factor (HCF) is 1. Example: (8, 15).
• Rational Numbers: Numbers that can be written in the form p/q, where p and q are integers and q ≠ 0. Example: 3/5, 4, -2/7.
• Irrational Numbers: Numbers that cannot be expressed as p/q. Their decimal expansions are non-terminating and non-repeating. Example: √2, √3, π.

2. Important Formulas, Rules, and Facts

• Sum of first n natural numbers: Sum = n × (n + 1) ÷ 2
• Sum of squares of first n natural numbers: Sum = n × (n + 1) × (2n + 1) ÷ 6
• Sum of cubes of first n natural numbers: Sum = [n × (n + 1) ÷ 2]²
• Sum of first n odd numbers: Sum = n²
• Sum of first n even numbers: Sum = n × (n + 1)

Divisibility Rules:

• Divisibility by 2: Last digit must be even (0, 2, 4, 6, 8).
• Divisibility by 3: Sum of digits must be divisible by 3.
• Divisibility by 4: Number formed by the last two digits must be divisible by 4.
• Divisibility by 5: Last digit must be 0 or 5.
• Divisibility by 8: Number formed by the last three digits must be divisible by 8.
• Divisibility by 9: Sum of digits must be divisible by 9.
• Divisibility by 11: The difference between the sum of digits at odd places and the sum of digits at even places must be either 0 or a multiple of 11.

Unit Digit Concept (Cyclicity):

• Cyclicity of 1: The unit digit of numbers ending in 0, 1, 5, 6 remains the same for any power. (e.g., 5^k always ends in 5).
• Cyclicity of 2: Numbers ending in 4 and 9 repeat their unit digit in a cycle of 2.
• 4^odd = 4, 4^even = 6
• 9^odd = 9, 9^even = 1
• Cyclicity of 4: Numbers ending in 2, 3, 7, 8 repeat their unit digit in a cycle of 4. Divide the power by 4 and find the remainder 'r'. The unit digit will be (Unit digit of base)^r. If remainder is 0, use power 4.

3. Solved Examples

Example 1:

Question: Find the unit digit of (273^137) × (412^218).

Solution:

1. Break down the expression into individual unit digits and powers:

• For 273^137, the unit digit of the base is 3. The cyclicity of 3 is 4.
• Divide the power 137 by 4: 137 ÷ 4 leaves a remainder of 1.
• So, unit digit of 273^137 = 3¹ = 3.

1. For 412^218, the unit digit of the base is 2. The cyclicity of 2 is 4.

• Divide the power 218 by 4: 218 ÷ 4 leaves a remainder of 2.
• So, unit digit of 412^218 = 2² = 4.

1. Multiply the individual unit digits:

• 3 × 4 = 12.
• The unit digit of 12 is 2.

Answer: The unit digit is 2.

Example 2:

Question: If the 9-digit number 897359y7x is completely divisible by 72, find the value of (3x - y).

Solution:

1. Since the number is divisible by 72, it must be divisible by both 8 and 9 (co-prime factors of 72).
2. Divisibility by 8: The last three digits (y7x) must be divisible by 8.

• Let's test values for y and x. If y = 4, then 472 is divisible by 8 (472 ÷ 8 = 59). So, x = 2 when y = 4.

1. Divisibility by 9: The sum of all digits must be divisible by 9.

• Sum of digits = 8 + 9 + 7 + 3 + 5 + 9 + y + 7 + x = 48 + x + y.
• Substitute x = 2 and y = 4: Sum = 48 + 2 + 4 = 54.
• Since 54 is divisible by 9, this pair is correct.

1. Calculate (3x - y):

• 3(2) - 4 = 6 - 4 = 2.

Answer: 2.

Example 3:

Question: Find the remainder when (67^67 + 67) is divided by 68.

Solution:

1. Express the term using negative remainders:

• 67 when divided by 68 leaves a remainder of -1 (since 67 = 68 - 1).

1. Substitute this into the expression:

• (67^67 + 67) ÷ 68
• Remainder = [(-1)^67 + (-1)] ÷ 68

1. Simplify the powers:

• Since 67 is an odd power, (-1)^67 = -1.
• Remainder = -1 + (-1) = -2.

1. Convert negative remainder to positive:

• Positive Remainder = Divisor + Negative Remainder
• Remainder = 68 - 2 = 66.

Answer: 66.

4. Shortcuts and Tricks

Trick 1: Finding the Number of Zeros at the end of N! (Factorial)

To find the number of trailing zeros in N!, divide N successively by 5 and add the quotients.

• Formula: Number of zeros = [N ÷ 5] + [N ÷ 25] + [N ÷ 125] + ... (where [ ] represents the integer quotient).
• Example: Number of zeros in 100! = (100 ÷ 5) + (100 ÷ 25) = 20 + 4 = 24 zeros.

Trick 2: Algebraic Divisibility Rules

• (a^n + b^n) is always divisible by (a + b) if n is odd.
• (a^n - b^n) is always divisible by (a - b) if n is odd or even.
• (a^n - b^n) is always divisible by (a + b) if n is even.

Trick 3: Sum of AP (Arithmetic Progression) for Number Series

If you need to find the sum of numbers with a common difference:

• Sum = (First term + Last term) × Number of terms ÷ 2
• Number of terms (n) = [(Last term - First term) ÷ Common Difference] + 1

5. Common Mistakes to Avoid

• Mistaking 1 as a Prime Number: 1 is neither prime nor composite because it has only one factor (itself). The smallest prime number is 2.
• Cyclicity Remainder of 0: When finding the unit digit, if the power is completely divisible by 4 (remainder = 0), do not use power 0. Use power 4. (e.g., Unit digit of 2^40 is 2^4 = 16 -> 6, not 2^0 = 1).
• Negative Remainder Confusion: Always remember that final remainders in exam options are positive. If you get a negative remainder like -3 with divisor 7, the actual remainder is 7 - 3 = 4.