Number System: Complete Master Guide for UPSC/MPSC CSAT (English)

Introduction to Number System in CSAT

The Number System is the foundation of mathematics and arguably the most important topic in the UPSC and MPSC CSAT syllabus. It deals with the classification, properties, and relationships of numbers. From basic arithmetic operations to advanced concepts like remainders and units digits, a strong grasp of the Number System is essential for solving quantitative aptitude questions efficiently.

In recent years, the UPSC has significantly increased the weightage and complexity of Number System questions. Understanding the underlying concepts rather than just memorizing formulas is crucial to tackling these tricky questions.

Importance in UPSC and MPSC

The Number System consistently accounts for 10-15 questions in the UPSC CSAT paper and holds a substantial share in MPSC as well. Questions often revolve around divisibility rules, remainders, prime numbers, LCM, HCF, and digits. Mastering this topic not only secures a large chunk of marks but also builds a solid foundation for other topics like Averages, Percentages, and Time & Work.

Core Concepts, Formulas, and Tricks

1. Classification of Numbers

• Natural Numbers (N): Counting numbers {1, 2, 3, …}
• Whole Numbers (W): Natural numbers including zero {0, 1, 2, 3, …}
• Integers (Z): All positive and negative whole numbers {…, -2, -1, 0, 1, 2, …}
• Rational Numbers (Q): Numbers that can be expressed as p/q, where p and q are integers and q ≠ 0.
• Prime Numbers: Numbers greater than 1 that have exactly two factors: 1 and themselves (e.g., 2, 3, 5, 7, 11). Note: 2 is the only even prime number. 1 is neither prime nor composite.
• Composite Numbers: Numbers that have more than two factors (e.g., 4, 6, 8, 9).

2. Divisibility Rules

• By 2: Last digit is 0, 2, 4, 6, or 8.
• By 3: Sum of the digits is divisible by 3.
• By 4: Number formed by the last two digits is divisible by 4.
• By 5: Last digit is 0 or 5.
• By 8: Number formed by the last three digits is divisible by 8.
• By 9: Sum of the digits is divisible by 9.
• By 11: The difference between the sum of digits at odd positions and even positions is either 0 or divisible by 11.

3. Unit Digit Concept (Cyclicity)

To find the unit digit of an expression like $X^N$, look at the unit digit of the base and the power cycle (cyclicity).

• Numbers ending in 0, 1, 5, 6 have cyclicity of 1 (unit digit remains the same).
• Numbers ending in 4, 9 have cyclicity of 2 (e.g., $4^1=4, 4^2=16$ (ends in 6), $4^3=64$ (ends in 4)).
• Numbers ending in 2, 3, 7, 8 have cyclicity of 4. Divide the power by 4 and find the remainder to determine the unit digit.

4. Remainder Theorem

If you need to find the remainder of $(A \times B \times C) / N$, it is equivalent to finding the remainders of $A/N$, $B/N$, $C/N$ individually, multiplying them, and then dividing by $N$ again.

Solved Examples with Step-by-Step Explanations

Example 1: Divisibility Rule

Question: If the number 738A6A is divisible by 11, find the value of A.

Step-by-Step Solution:

• Rule for 11: (Sum of digits at odd positions) – (Sum of digits at even positions) = 0 or multiple of 11.
• Odd positions (from left): 7, 8, 6. Sum = $7 + 8 + 6 = 21$.
• Even positions (from left): 3, A, A. Sum = $3 + A + A = 3 + 2A$.
• Difference = $21 – (3 + 2A) = 18 – 2A$.
• For $18 – 2A$ to be divisible by 11 or be 0:
  • If $18 – 2A = 0 \Rightarrow 2A = 18 \Rightarrow A = 9$.
  • Let’s check: $18 – 2(9) = 0$. Since 0 is divisible by 11, this works.
• Answer: A = 9

Example 2: Unit Digit Concept

Question: Find the unit digit of $237^{153} \times 164^{72}$.

Step-by-Step Solution:

• Part 1: $237^{153}$
  • Unit digit of base is 7. Cyclicity of 7 is 4.
  • Divide power 153 by 4. $153 = 4 \times 38 + 1$. Remainder is 1.
  • So, unit digit is $7^1 = 7$.
• Part 2: $164^{72}$
  • Unit digit of base is 4. For 4, if power is even, unit digit is 6. (Since 72 is even).
• Multiply the unit digits: $7 \times 6 = 42$.
• The unit digit of the product is 2.
• Answer: 2

Example 3: Finding Remainders

Question: What is the remainder when $15^{40}$ is divided by 14?

Step-by-Step Solution:

• We can write $15$ as $(14 + 1)$.
• So, $(14 + 1)^{40} / 14$.
• Using the binomial theorem/remainder concept, any term containing 14 will be divisible by 14.
• The remainder will be strictly determined by $1^{40}$.
• $1^{40} = 1$.
• Answer: 1

Pro-Tips to Avoid Common Mistakes

• Read carefully: ‘Whole numbers’ include 0, ‘Natural numbers’ start from 1. 1 is NOT a prime number. 2 is the ONLY EVEN prime number. These small facts often form the catch in statement-based questions.
• Zero is even: Remember that 0 is an even integer.
• Negative remainders: For quick calculations, use negative remainders. E.g., when 13 is divided by 14, remainder is 13, but you can also treat it as -1. $(-1)^{\text{even}} = 1$.
• Don’t brute force: If a calculation seems too long (like large powers), stop and look for cyclicity, patterns, or divisibility rules. UPSC never tests brute calculation speed; it tests conceptual clarity.

Practice Questions

Try these out to solidify your concepts:

1. How many prime numbers are there between 1 and 50?
2. Find the unit digit of $3^{41} \times 7^{42} \times 8^{43}$.
3. What is the remainder when $17^{200}$ is divided by 18?
4. If $746832X71$ is divisible by 9, find the value of X.
5. What is the sum of the first 100 natural numbers?

Answers:

• 1) 15 (They are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47)
• 2) Unit digit of $3^{41} \rightarrow$ rem(41/4)=1 $\rightarrow 3^1=3$. Unit digit of $7^{42} \rightarrow$ rem(42/4)=2 $\rightarrow 7^2=9$. Unit digit of $8^{43} \rightarrow$ rem(43/4)=3 $\rightarrow 8^3=2$. So, $3 \times 9 \times 2 = 54$. Unit digit is 4.
• 3) $17 \equiv -1$ (mod 18). So, $(-1)^{200} = 1$. Remainder is 1.
• 4) Sum of digits = $7+4+6+8+3+2+X+7+1 = 38 + X$. For this to be divisible by 9, the next multiple is 45. So, $X = 45 – 38 = 7$.
• 5) Sum = $n(n+1)/2 = 100(101)/2 = 5050$.