This course aims to develop students' analytical skills by applying procedural fluency with number systems and mathematical operations to solve problems related to percentage, profit & loss, data series, ratio, proportion, permutation, combination, and reasoning.
Review the core definitions, standard architectures, and syntax patterns below. Pay special attention to sections highlighted with bullet points for your mid-term and end-term subjective questions.
Unit 1: Vedic Mathematics, Number Systems, and Averages
1. Vedic Mathematics and Fast Mental Algorithms
Vedic Mathematics is a system of mental arithmetic reconstructed between 1911 and 1918 by Swami Bharati Krishna Tirtha from the Atharva Veda. The architecture unifies arithmetic, algebraic, and geometric operations into sixteen fundamental Sutras (word-formulae) and thirteen Upa-Sutras (sub-formulae).
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| Vedic Computational Architecture |
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v v v
[ Base Method: Nikhilam ] [ Crosswise: Urdhva-Tiryag ] [ Duplex: Dwandwa Yoga ]
All from 9 & Last from 10 Vertical & Cross Multiplic. Mental Squaring & Roots
1.1 Core Vedic Sutras for Rapid Arithmetic
1. Nikhilam Navatashcaramam Dashatah ("All from 9 and the last from 10")
This sutra facilitates rapid multiplication and subtraction when numbers lie in proximity to powers of 10 (10,100,1000,10k), known as the Working Base.
Case 1: Both Operands Below the Base (Base=100)
Let numbers be A=100−d1 and B=100−d2, where d1,d2 are negative deviations:
96 \quad (-04) \\
\times 93 \quad (-07) \\
\hline
\text{Left Part} &= 96 - 07 = 93 - 04 = 89 \\
\text{Right Part} &= (-04) \times (-07) = 28 \\
\text{Result} &= 8928
\end{aligned}$$
*Rule:* The right-hand part must contain exactly as many digits as there are zeros in the base ($100 \implies 2 \text{ digits}$).
Case 2: Both Operands Above the Base (Base=100)
Let A=100+d1 and B=100+d2:
108 \quad (+08) \\
\times 105 \quad (+05) \\
\hline
\text{Left Part} &= 108 + 05 = 113 \\
\text{Right Part} &= (+08) \times (+05) = 40 \\
\text{Result} &= 11340
\end{aligned}$$
2. Urdhva-Tiryagbhyam ("Vertically and Crosswise")
The master universal multiplication algorithm applicable to numbers of any arbitrary length, completely independent of proximity to any base.
The 2-Digit Algorithm (ab×cd):
\text{Step 1 (Vertical Right):} \quad & b \times d \\
\text{Step 2 (Crosswise):} \quad & (a \times d) + (b \times c) \\
\text{Step 3 (Vertical Left):} \quad & a \times c
\end{aligned}$$
*Example: Multiply $43 \times 27$*
1. Vertical Right: $3 \times 7 = 21$ (Write 1, carry 2).
2. Crosswise: $(4 \times 7) + (3 \times 2) = 28 + 6 = 34$; add carry 2 $\implies 36$ (Write 6, carry 3).
3. Vertical Left: $4 \times 2 = 8$; add carry 3 $\implies 11$.
4. Result: $1161$.
##### The 3-Digit Algorithm ($abc \times def$):
$$\begin{matrix}
\text{Step 1:} & c \times f \\
\text{Step 2:} & b f + c e \\
\text{Step 3:} & a f + c d + b e \\
\text{Step 4:} & a e + b d \\
\text{Step 5:} & a d
\end{matrix}$$
---
#### 3. Ekadhikena Purvena ("By One More Than the Previous One")
Enables instantaneous mental squaring of any positive integer ending in the digit 5:
$$(N5)^2 = [N \times (N + 1)] \ || \ 25$$
- For $65^2$: Left part $= 6 \times (6 + 1) = 6 \times 7 = 42$; Right part $= 25 \implies 4225$.
- For $115^2$: Left part $= 11 \times 12 = 132$; Right part $= 25 \implies 13225$.
- Also applies to general multiplication $A \times B$ when initial digits are identical and the sum of final units digits equals 10:
$$43 \times 47 = [4 \times 5] \ || \ [3 \times 7] = 2021$$
---
#### 4. The Duplex Method (Dwandwa Yoga) for Squaring
The Duplex ($D$) of a sequence of digits is defined as:
- For a single digit $a$: $D(a) = a^2$
- For a 2-digit number $ab$: $D(ab) = 2ab$
- For a 3-digit number $abc$: $D(abc) = 2ac + b^2$
- For a 4-digit number $abcd$: $D(abcd) = 2ad + 2bc$
- For a 5-digit number $abcde$: $D(abcde) = 2ae + 2bd + c^2$
The square of any multi-digit number is formed by concatenating duplexes from left to right:
$$(abc)^2 = D(a) \ || \ D(ab) \ || \ D(abc) \ || \ D(bc) \ || \ D(c)$$
*Example: Find $314^2$*
- $D(3) = 3^2 = 9$
- $D(31) = 2(3)(1) = 6$
- $D(314) = 2(3)(4) + 1^2 = 24 + 1 = 25$
- $D(14) = 2(1)(4) = 8$
- $D(4) = 4^2 = 16$
Concatenation with carries:
$$9 \ || \ 6 \ || \ 25 \ || \ 8 \ || \ 16 = 9 \ || \ 6 \ || \ 25 \ || \ 9 \ || \ 6 = 9 \ || \ 8 \ || \ 5 \ || \ 9 \ || \ 6 = 98596$$
---
## 2. Theoretical Number Systems and Modular Arithmetic
### 2.1 The Real Number Hierarchy
$$\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$$
- **Natural Numbers ($\mathbb{N}$):** Counting numbers $\{1, 2, 3, \dots\}$.
- **Whole Numbers ($\mathbb{W}$):** Natural numbers including zero $\{0, 1, 2, \dots\}$.
- **Integers ($\mathbb{Z}$):** $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.
- **Rational Numbers ($\mathbb{Q}$):** Numbers expressible as $\frac{p}{q}$ ($p, q \in \mathbb{Z}, q \ne 0$). Characterized by terminating or recurring infinite decimal expansions.
- **Irrational Numbers ($\mathbb{Q}^c$):** Real numbers that cannot be expressed as ratios of integers (e.g., $\sqrt{2}, \pi, e$). Their decimal expansions are non-terminating and non-recurring.
---
### 2.2 Prime Numbers and the Fundamental Theorem of Arithmetic
#### Theorem 2.1: The Fundamental Theorem of Arithmetic
Every positive integer $n > 1$ can be represented uniquely as a product of prime powers, up to the order of factors:
$$n = p_1^{a_1} \cdot p_2^{a_2} \cdots p_k^{a_k} = \prod_{i=1}^k p_i^{a_i}$$
where $p_1 < p_2 < \dots < p_k$ are distinct prime numbers, and $a_i \ge 1$ are positive integer exponents.
#### Factor Counting and Factor Functions:
From the canonical prime factorization $n = \prod_{i=1}^k p_i^{a_i}$:
1. **Total Number of Factors ($\tau(n)$ or $d(n)$):**
$$\tau(n) = (a_1 + 1)(a_2 + 1)\cdots(a_k + 1) = \prod_{i=1}^k (a_i + 1)$$
2. **Sum of Factors ($\sigma(n)$):**
$$\sigma(n) = \prod_{i=1}^k \left( \frac{p_i^{a_i + 1} - 1}{p_i - 1} \right) = (1 + p_1 + \dots + p_1^{a_1})\cdots(1 + p_k + \dots + p_k^{a_k})$$
3. **Product of Factors ($P(n)$):**
$$P(n) = n^{\tau(n) / 2}$$
4. **Number of Odd Factors:** Set $a_1 = 0$ for $p_1 = 2$:
$$\tau_{\text{odd}}(n) = \prod_{i=2}^k (a_i + 1)$$
5. **Number of Even Factors:**
$$\tau_{\text{even}}(n) = \tau(n) - \tau_{\text{odd}}(n) = a_1 \prod_{i=2}^k (a_i + 1)$$
---
### 2.3 Comprehensive Divisibility Rules
| Divisor | Mathematical Condition |
| :--- | :--- |
| **2** | Last digit is even ($0, 2, 4, 6, 8$). |
| **3** | Digital root (sum of digits) is divisible by 3. |
| **4** | Integer formed by the last two digits is divisible by 4. |
| **5** | Last digit is 0 or 5. |
| **6** | Divisible simultaneously by both 2 and 3. |
| **7** | Double the units digit and subtract from the remaining truncated number; difference is divisible by 7. |
| **8** | Integer formed by the last three digits is divisible by 8. |
| **9** | Sum of digits is divisible by 9. |
| **10** | Last digit is 0. |
| **11** | Alternating sum of digits: $|(S_{\text{odd}} - S_{\text{even}})|$ is 0 or a multiple of 11. |
| **12** | Divisible simultaneously by 3 and 4 (coprime factors). |
| **13** | Multiply units digit by 4 and add to truncated number; result is divisible by 13. |
| **17** | Multiply units digit by 5 and subtract from truncated number; result is divisible by 17. |
| **19** | Multiply units digit by 2 and add to truncated number; result is divisible by 19. |
---
### 2.4 Cyclicity and Unit Digit Computations
The unit digit of an exponential expression $a^b$ depends exclusively on the unit digit of $a$ and follows a periodic cycle of length at most 4:
| Last Digit of $a$ | Power Cycle Pattern | Periodicity (Cycle Length) |
| :--- | :--- | :--- |
| **0, 1, 5, 6** | $[0], [1], [5], [6]$ (Fixed values) | **1** |
| **4** | $4^1 = 4, 4^2 = 6, 4^3 = 4, \dots$ | **2** ($4$ if odd, $6$ if even) |
| **9** | $9^1 = 9, 9^2 = 1, 9^3 = 9, \dots$ | **2** ($9$ if odd, $1$ if even) |
| **2** | $2, 4, 8, 6, 2, \dots$ | **4** |
| **3** | $3, 9, 7, 1, 3, \dots$ | **4** |
| **7** | $7, 9, 3, 1, 7, \dots$ | **4** |
| **8** | $8, 4, 2, 6, 8, \dots$ | **4** |
#### General Algorithm for Unit Digit of $a^b$:
1. Identify unit digit $u = a \pmod{10}$.
2. If $u \in \{0, 1, 5, 6\}$, unit digit is $u$.
3. Otherwise, divide exponent $b$ by the cyclicity $C$ (where $C = 4$ works universally for all digits):
$$r = b \pmod 4$$
- If $r > 0$, unit digit is $u^r \pmod{10}$.
- If $r = 0$, unit digit is $u^4 \pmod{10}$.
---
### 2.5 Highest Power of a Prime in Factorials (Legendre's Formula)
#### Theorem 2.2: Legendre's Formula
The exact exponent $E_p(n!)$ of a prime number $p$ in the prime factorization of $n!$ is:
$$E_p(n!) = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor = \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + \cdots$$
The series terminates once $p^k > n$.
#### Application: Trailing Zeros in $n!$
A trailing zero is generated by the product $2 \times 5 = 10$. In $n!$, factors of 2 are always strictly more abundant than factors of 5. Consequently, the number of trailing zeros in $n!$ equals the highest power of 5 dividing $n!$:
$$\text{Trailing Zeros in } n! = E_5(n!) = \left\lfloor \frac{n}{5} \right\rfloor + \left\lfloor \frac{n}{25} \right\rfloor + \left\lfloor \frac{n}{125} \right\rfloor + \cdots$$
*Example: Trailing zeros in $100!$*
$$E_5(100!) = \left\lfloor \frac{100}{5} \right\rfloor + \left\lfloor \frac{100}{25} \right\rfloor + \left\lfloor \frac{100}{125} \right\rfloor = 20 + 4 + 0 = 24 \text{ zeros}$$
---
### 2.6 Advanced Remainder Theorems
1. **Fermat's Little Theorem:**
If $p$ is a prime number and $\gcd(a, p) = 1$, then:
$$a^{p - 1} \equiv 1 \pmod p \implies a^p \equiv a \pmod p$$
2. **Euler's Totient Theorem:**
For any modulus $m \in \mathbb{N}$ and integer $a$ coprime to $m$ ($\gcd(a, m) = 1$):
$$a^{\phi(m)} \equiv 1 \pmod m$$
where Euler's totient function $\phi(m) = m \prod_{p \mid m} \left(1 - \frac{1}{p}\right)$ counts integers in $\{1, \dots, m\}$ coprime to $m$.
3. **Wilson's Theorem:**
An integer $p > 1$ is prime if and only if:
$$(p - 1)! \equiv -1 \pmod p \quad (\text{or } (p - 1)! + 1 \equiv 0 \pmod p)$$
---
### 2.7 HCF and LCM Theorems
- **Product Identity:** For any two positive integers $A$ and $B$:
$$A \times B = \text{HCF}(A, B) \times \text{LCM}(A, B)$$
- **Fractions Rule:**
$$\text{HCF}\left( \frac{a}{b}, \frac{c}{d} \right) = \frac{\text{HCF}(a, c)}{\text{LCM}(b, d)}$$
$$\text{LCM}\left( \frac{a}{b}, \frac{c}{d} \right) = \frac{\text{LCM}(a, c)}{\text{HCF}(b, d)}$$
---
## 3. Mathematical Foundations of Averages
### 3.1 Definitions and Fundamental Principles
#### 1. Arithmetic Mean
For a set of $n$ observations $x_1, x_2, \dots, x_n$:
$$\bar{X} = \frac{\sum_{i=1}^n x_i}{n} \implies \sum_{i=1}^n x_i = n \cdot \bar{X}$$
#### 2. Weighted Arithmetic Mean
When observations $x_i$ carry unequal relative importances or frequencies $w_i$:
$$\bar{X}_w = \frac{\sum_{i=1}^k w_i x_i}{\sum_{i=1}^k w_i}$$
#### 3. Assumed Mean (Deviation) Method
Let $A$ be an assumed mean, and $d_i = x_i - A$ be deviations:
$$\bar{X} = A + \frac{\sum_{i=1}^n d_i}{n}$$
*Key Invariant:* The algebraic sum of deviations from the true arithmetic mean is identically zero:
$$\sum_{i=1}^n (x_i - \bar{X}) = 0$$
---
### 3.2 Dynamic Averages: Inclusion, Exclusion, and Replacement
#### 1. Addition (Inclusion) of a New Member
If the average of $n$ members is $A$, and the addition of a new member increases the average to $A'$:
$$\text{Value of New Member} = A + (n + 1)(A' - A)$$
If the addition decreases the average:
$$\text{Value of New Member} = A - (n + 1)(A - A')$$
#### 2. Removal (Exclusion) of a Member
If excluding one member from a group of $n$ alters the average from $A$ to $A'$:
$$\text{Value of Excluded Member} = A + (n - 1)(A - A')$$
#### 3. Replacement of an Existing Member
If one person of weight/value $W_{\text{old}}$ is replaced by a newcomer among $n$ members, shifting the average by $\Delta A$:
$$W_{\text{new}} = W_{\text{old}} + n \cdot (\Delta A)$$
where $\Delta A > 0$ for an average increase, and $\Delta A < 0$ for a decrease.
---
### 3.3 Averages of Standard Sequences
1. **First $n$ Natural Numbers:**
$$\text{Sum} = \frac{n(n + 1)}{2} \implies \text{Average} = \frac{n + 1}{2}$$
2. **First $n$ Even Natural Numbers ($2, 4, 6, \dots, 2n$):**
$$\text{Sum} = n(n + 1) \implies \text{Average} = n + 1$$
3. **First $n$ Odd Natural Numbers ($1, 3, 5, \dots, 2n - 1$):**
$$\text{Sum} = n^2 \implies \text{Average} = n$$
4. **Squares of First $n$ Natural Numbers ($1^2, 2^2, \dots, n^2$):**
$$\text{Sum} = \frac{n(n + 1)(2n + 1)}{6} \implies \text{Average} = \frac{(n + 1)(2n + 1)}{6}$$
5. **Cubes of First $n$ Natural Numbers ($1^3, 2^3, \dots, n^3$):**
$$\text{Sum} = \left[ \frac{n(n + 1)}{2} \right]^2 \implies \text{Average} = \frac{n(n + 1)^2}{4}$$
6. **Average of Any Arithmetic Progression (AP):**
$$\text{Average} = \frac{\text{First Term} + \text{Last Term}}{2}$$
---
### 3.4 Harmonic Mean and Average Speed
When traveling equal distances $D$ at different speeds $v_1$ and $v_2$:
$$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2D}{\frac{D}{v_1} + \frac{D}{v_2}} = \frac{2 v_1 v_2}{v_1 + v_2}$$
*Crucial Principle:* Average speed over equal distances is the **Harmonic Mean** of speeds, never the simple arithmetic mean!
If traveling for equal time intervals $T$:
$$\text{Average Speed} = \frac{v_1 T + v_2 T}{2T} = \frac{v_1 + v_2}{2} \quad (\text{Arithmetic Mean})$$