Number System

Number System

Overview

The Number System forms the bedrock of Quantitative Ability in HSSC CET. Nearly every arithmetic topic—from simplification to data interpretation—relies on your fluency with numbers, their types, and their properties.

Mastering this topic gives you speed. When you instinctively know that 72 is divisible by 8 (because 72 = 8 × 9), or that the unit digit of 7⁴ is 1, you save precious seconds. Focus on: classification of numbers, divisibility rules for 2–11, place value concepts, and remainder tricks. These are your exam weapons.

Key Concepts

• Natural Numbers (N): Counting numbers starting from 1. Example: 1, 2, 3, 4, ...
• Whole Numbers (W): Natural numbers plus zero. Example: 0, 1, 2, 3, ...
• Integers (Z): All whole numbers and their negatives. Example: ..., −3, −2, −1, 0, 1, 2, 3, ...
• Rational Numbers: Numbers expressible as p/q where p, q are integers and q ≠ 0. Every terminating or repeating decimal is rational.
• Place Value vs Face Value: In 5832, the face value of 8 is simply 8, but the place value of 8 is 800 (8 × 100).
• Even and Odd Numbers: Even numbers are divisible by 2 (end in 0, 2, 4, 6, 8); odd numbers are not.
• Prime Numbers: Numbers greater than 1 with exactly two factors (1 and itself). First ten primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
• Composite Numbers: Numbers with more than two factors. Note: 1 is neither prime nor composite.

Formulas / Key Facts

Divisibility Rules (memorise these cold):

Unit Digit Cyclicity:

• 2 → cycle of 4: (2, 4, 8, 6)
• 3 → cycle of 4: (3, 9, 7, 1)
• 7 → cycle of 4: (7, 9, 3, 1)
• 8 → cycle of 4: (8, 4, 2, 6)
• 4 → cycle of 2: (4, 6)
• 9 → cycle of 2: (9, 1)
• 0, 1, 5, 6 → always end in themselves

Number of Factors Formula: If N = a^p × b^q × c^r, then total factors = (p+1)(q+1)(r+1)

Sum of First n Natural Numbers: n(n+1)/2

Sum of First n Natural Number Squares: n(n+1)(2n+1)/6

Worked Examples

Example 1: Divisibility Test

Question: Is 4,73,256 divisible by 8?

Solution:

• For divisibility by 8, check last three digits: 256
• 256 ÷ 8 = 32 (exact division)
• Answer: Yes, divisible by 8

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Example 2: Place Value Application

Question: Find the difference between the place value and face value of 7 in 9,07,452.

Solution:

• Position of 7: thousands place
• Place value of 7 = 7 × 1000 = 7000
• Face value of 7 = 7
• Difference = 7000 − 7 = 6993

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Example 3: Unit Digit

Question: Find the unit digit of 7^243.

Solution:

• Cyclicity of 7 is 4 (unit digits repeat: 7, 9, 3, 1)
• Divide exponent by 4: 243 ÷ 4 = 60 remainder 3
• Remainder 3 means third position in cycle: 7, 9, 3, 1
• Answer: 3

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Example 4: Number of Factors

Question: How many factors does 360 have?

Solution:

• Prime factorisation: 360 = 2³ × 3² × 5¹
• Number of factors = (3+1)(2+1)(1+1) = 4 × 3 × 2 = 24 factors

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Example 5: Divisibility by 11

Question: Check if 918,082 is divisible by 11.

Solution:

• Digits from right: 2, 8, 0, 8, 1, 9
• Odd positions (1st, 3rd, 5th from right): 2 + 0 + 1 = 3
• Even positions (2nd, 4th, 6th from right): 8 + 8 + 9 = 25
• Difference: |3 − 25| = 22
• 22 is divisible by 11
• Answer: Yes, divisible by 11

Common Mistakes

• Confusing place value with face value → Remember: place value depends on position (hundreds, thousands, etc.), face value is just the digit itself. In 4052, place value of 5 is 50, not 5.
• Forgetting that 1 is not prime → Prime numbers must have exactly two distinct factors. The number 1 has only one factor (itself), so it is neither prime nor composite.
• Applying the wrong divisibility rule for 4 vs 8 → For 4, check last two digits. For 8, check last three digits. Students often mix these up under pressure.
• Miscounting cyclicity remainder → When remainder is 0, use the last element of the cycle, not the first. For 2^8, remainder of 8÷4 = 0, so unit digit is 6 (the fourth element), not 2.
• Adding instead of subtracting for divisibility by 11 → You must find the difference between sum of alternate digits, not their total sum.
• Assuming all odd numbers are prime → 9, 15, 21, 25, etc., are odd but composite. Always verify by checking for factors.

Quick Reference

• Divisibility by 6 = divisible by BOTH 2 and 3.
• Sum of digits trick: quickly add digits to test divisibility by 3 or 9.
• Unit digit cycles: 2, 3, 7, 8 repeat every 4 powers; 4, 9 repeat every 2 powers.
• Factors formula: N = a^p × b^q → Total factors = (p+1)(q+1).
• First 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 (only even prime is 2).
• Place value = digit × position value (ones = 1, tens = 10, hundreds = 100, etc.).