2.1 Number Systems, Divisibility Rules, Prime Factorization & Remainder Theorem

Mathematical practice context

The numerical railway scenarios in this guide are fictional aptitude exercises. They use the stated idealized assumptions and do not specify actual operating rules, equipment performance, tariffs, staffing norms or engineering designs.

Classification of Numbers & Set Topology

The real number system $\mathbb{R}$ comprises all values representing continuous quantities along the number line. It divides into two mutually exclusive sets: rational numbers ($\mathbb{Q}$) and irrational numbers ($\mathbb{Q}'$).

Number Sets & Definitions

1. Natural Numbers ($\mathbb{N}$): Counting numbers $\{1, 2, 3, 4, \dots\}$. Natural numbers are used for discrete tallies such as the number of passenger coaches in a rake or signals in a block section.
2. Whole Numbers ($\mathbb{W}$): Natural numbers including zero $\{0, 1, 2, 3, \dots\}$. Zero represents baseline null states, such as zero block failures during a controller's 8-hour shift.
3. Integers ($\mathbb{Z}$): Positive and negative whole numbers along with zero $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$. Integers express timetable deviations, where $+4$ indicates minutes ahead of schedule and $-7$ denotes minutes of operational delay.
4. Rational Numbers ($\mathbb{Q}$): Any number expressible in the fractional form $\frac{p}{q}$, where $p, q \in \mathbb{Z}$ and $q \neq 0$. In decimal form, rational numbers either terminate (e.g., $\frac{5}{8} = 0.625$) or form an infinitely repeating periodic sequence (e.g., $\frac{4}{11} = 0.\overline{36}$).
5. Irrational Numbers ($\mathbb{Q}'$): Real numbers that cannot be represented as a ratio of integers. Their decimal expansions are non-terminating and non-repeating (e.g., $\sqrt{2}, \sqrt{3}, \pi$). For example, a square with unit-length sides has an irrational diagonal length $\sqrt{2}$.
6. Real Numbers ($\mathbb{R}$): The union of all rational and irrational numbers ($\mathbb{R} = \mathbb{Q} \cup \mathbb{Q}'$).
7. Prime Numbers: Integers greater than 1 that possess exactly two distinct positive divisors: 1 and the number itself. The integer 2 is the smallest prime number and the only even prime. The number 1 is neither prime nor composite.
8. Composite Numbers: Integers greater than 1 possessing more than two distinct positive divisors. The smallest composite integer is 4.
9. Co-prime Numbers (Relatively Prime): Any pair of integers $a$ and $b$ whose greatest common divisor is 1, denoted as $\gcd(a, b) = 1$. The individual numbers need not be primes (e.g., 8 and 15 are co-prime because $\gcd(8, 15) = 1$, even though both are composite).
10. Twin Primes: Pairs of prime numbers that differ by exactly 2, such as $(3, 5), (5, 7), (11, 13), (17, 19), (29, 31),$ and $(41, 43)$.

---

Primality Testing Algorithm

First reject every integer $N\le1$ as nonprime. For an integer $N>1$, test primality without exhaustive trial division as follows:

1. Find the integer floor of the square root of $N$: $K = \lfloor\sqrt{N}\rfloor$.
2. Test divisibility of $N$ only by prime numbers less than or equal to $K$.
3. For an integer $N > 1$, if no prime $p \le K$ divides $N$, then $N$ is prime. Zero, one and negative integers are not primes.

+-----------------------------------------------------------------------------+
|                         PRIMALITY TESTING FLOWCHART                         |
+-----------------------------------------------------------------------------+
| Given Integer N > 1                                                             |
|    │                                                                        |
|    ▼                                                                        |
| Calculate K = floor(sqrt(N))                                                |
|    │                                                                        |
|    ▼                                                                        |
| List all prime numbers p <= K                                               |
|    │                                                                        |
|    ├──> Does any prime p divide N evenly (N mod p == 0)?                    |
|    │       │                                                                |
|    │       ├──> YES: N is COMPOSITE                                         |
|    │       │                                                                |
|    │       └──> NO:  N is PRIME                                             |
+-----------------------------------------------------------------------------+

---

Divisibility Rules from 2 to 19

Divisibility tests help check options and factor expressions without performing full long division.

In-Depth Modular Mechanisms for Prime Divisors

For any two-part split of integer $N = 10a + b$, where $b$ is the units digit and $a$ is the truncated leading number:

• Divisibility by 7: $10a + b \equiv 0 \pmod 7 \iff 3a + b \equiv 0 \iff -4a + b \equiv 0 \iff a - 2b \equiv 0 \pmod 7$. Example: Test $2,429$: $242 - 2(9) = 224$; next step: $22 - 2(4) = 14$. Since $14$ is divisible by 7, $2,429$ is divisible by 7.
• Divisibility by 13: $10a + b \equiv 0 \pmod{13} \iff -3a + b \equiv 0 \iff 9a - 3b \equiv 0 \iff a + 4b \equiv 0 \pmod{13}$. Example: Test $1,014$: $101 + 4(4) = 117$; next step: $11 + 4(7) = 39$. Since $39 = 13 \times 3$, $1,014$ is divisible by 13.
• Divisibility by 17: $10a + b \equiv 0 \pmod{17} \iff 10a + b - 51b = 10(a - 5b) \equiv 0 \iff a - 5b \equiv 0 \pmod{17}$. Example: Test $2,261$: $226 - 5(1) = 221$; next step: $22 - 5(1) = 17$. Since 17 is divisible by 17, $2,261$ is divisible by 17.
• Divisibility by 19: $10a + b \equiv 0 \pmod{19} \iff 10a + b + 19b = 10(a + 2b) \equiv 0 \iff a + 2b \equiv 0 \pmod{19}$. Example: Test $1,083$: $108 + 2(3) = 114$; next step: $11 + 2(4) = 19$. Since 19 is divisible by 19, $1,083$ is divisible by 19.

---

Compound Divisibility: 72, 88, and 99 Rules

To test divisibility by a composite $M$ using two factors whose product is $M$, choose $a,b$ such that $\gcd(a, b) = 1$.

Three Composite Divisor Examples

1. Divisibility by 72 ($8 \times 9$):
   • Apply the rule for 8: The number formed by the last three digits must be divisible by 8.
   • Apply the rule for 9: The sum of all digits must be divisible by 9.
2. Divisibility by 88 ($8 \times 11$):
   • Apply the rule for 8: The last three digits must form a number divisible by 8.
   • Apply the rule for 11: The alternating digital sum $|\sum_{\text{odd}} - \sum_{\text{even}}| \equiv 0 \pmod{11}$.
3. Divisibility by 99 ($9 \times 11$):
   • Apply the rule for 9: The total sum of digits must be divisible by 9.
   • Apply the rule for 11: The alternating digital sum must be 0 or a multiple of 11.

---

Canonical Prime Factorization & Factor Functions

By the Fundamental Theorem of Arithmetic, every integer $N > 1$ can be uniquely factored into prime powers:

$$N = p_1^{a} \cdot p_2^{b} \cdot p_3^{c} \cdots p_k^{m}$$

where $p_1 < p_2 < \dots < p_k$ are distinct prime numbers, and $a, b, c, \dots, m$ are positive integers.

+-----------------------------------------------------------------------------+
|                     FACTOR COUNTING & SUMMATION FORMULAS                    |
+-----------------------------------------------------------------------------+
| Total Number of Factors:                                                    |
|    T(N) = (a + 1)(b + 1)(c + 1)...(m + 1)                                   |
|                                                                             |
| Number of Odd Factors (assuming p_1 = 2):                                   |
|    T_odd(N) = (b + 1)(c + 1)...(m + 1)                                      |
|                                                                             |
| Number of Even Factors:                                                     |
|    T_even(N) = a(b + 1)(c + 1)...(m + 1) = T(N) - T_odd(N)                  |
|                                                                             |
| Sum of All Factors:                                                         |
|    S(N) = [(p_1^(a+1) - 1)/(p_1 - 1)] * [(p_2^(b+1) - 1)/(p_2 - 1)] * ...   |
|                                                                             |
| Product of All Factors:                                                     |
|    P(N) = N^(T(N) / 2)                                                      |
+-----------------------------------------------------------------------------+

Factor Structure Analysis for $N = 2^a \cdot p_2^b \cdot p_3^c$

• Even Factors: An even factor must contain at least one factor of 2. Hence, the exponent of 2 can take $a$ values $\{1, 2, \dots, a\}$ instead of $(a+1)$ values. Therefore, $T_{\text{even}} = a(b+1)(c+1)$.
• Odd Factors: An odd factor cannot contain any factor of 2. The exponent of 2 must be 0 (1 choice). Therefore, $T_{\text{odd}} = (b+1)(c+1)$.
• Sum of Even Factors:

$$S_{\text{even}}(N) = (2^1 + 2^2 + \dots + 2^a) \times \left(\frac{p_2^{b+1}-1}{p_2-1}\right) \times \left(\frac{p_3^{c+1}-1}{p_3-1}\right)$$

• Sum of Odd Factors:

$$S_{\text{odd}}(N) = (2^0) \times \left(\frac{p_2^{b+1}-1}{p_2-1}\right) \times \left(\frac{p_3^{c+1}-1}{p_3-1}\right)$$

---

Remainder Theorems & Modular Arithmetic

Division Algorithm

For any integer dividend $N$ and positive divisor $D$, there exist unique integers $Q$ (quotient) and $R$ (remainder) such that:

$$N = D \times Q + R, \quad \text{where } 0 \le R < D$$

The Negative Remainder Method

For the displayed shortcut, choose a negative representative r satisfying -D < r < 0. If a representative lies outside this interval, reduce it modulo D first. When the dividend is just below a multiple of the divisor, modular calculations are dramatically simplified by using a negative remainder:

$$R_{\text{positive}} = D - |R_{\text{negative}}|$$

For example, $67 \div 68$ has a remainder of $67$, which is congruent to $-1 \pmod{68}$.

Fermat's Little Theorem (FLT)

If $p$ is a prime number and $a$ is an integer co-prime to $p$ ($\gcd(a, p) = 1$):

$$a^{p-1} \equiv 1 \pmod p$$

Consequently, to compute $a^n \pmod p$, reduce the exponent modulo $(p - 1)$:

$$a^n \equiv a^{n \bmod (p - 1)} \pmod p$$

Euler's Totient Theorem

For any modulus $m > 1$ and integer $a$ satisfying $\gcd(a, m) = 1$:

$$a^{\phi(m)} \equiv 1 \pmod m$$

where $\phi(m)$ is Euler's totient function, which counts integers in $\{1, 2, \dots, m\}$ co-prime to $m$. For prime-factorized $m = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$:

$$\phi(m) = m \left(1 - \frac{1}{p_1}\right)\left(1 - \frac{1}{p_2}\right)\cdots\left(1 - \frac{1}{p_r}\right)$$

Wilson's Theorem

If $p$ is a prime number, then:

$$(p - 1)! \equiv -1 \equiv p - 1 \pmod p$$

Dividing both sides by $(p - 1) \equiv -1 \pmod p$ gives the useful corollary:

$$(p - 2)! \equiv 1 \pmod p$$

Polynomial & Binomial Remainder Theorems

In the integer-divisor examples below, assume x is an integer greater than 1 and n is a positive integer.

1. $(x + 1)^n / x$: Leaves a remainder of $1^n = 1$ for all integer $n \ge 1$.
2. $(x - 1)^n / x$: Leaves a remainder of $(-1)^n$. If $n$ is even, remainder is $1$. If $n$ is odd, remainder is $-1 \equiv x - 1$.
3. Algebraic Factorization Properties:
   • $(a^n + b^n)$ is always divisible by $(a + b)$ whenever $n$ is odd.
   • $(a^n - b^n)$ is always divisible by $(a - b)$ for all natural numbers $n$.
   • $(a^n - b^n)$ is always divisible by both $(a - b)$ and $(a + b)$ whenever $n$ is even.

---

Worked Examples

Worked Example 1: Rolling Stock Identification Audit (Divisibility by 88)

Problem: A computerized railway automated wagon management system scans an 8-digit rolling stock serial code as $342x18y6$. In this fictional arithmetic exercise, the code is required to be divisible by 88. If $x$ and $y$ are single decimal digits, find the maximum possible value of $(3x + 2y)$.

• Step 1: Decompose 88 into co-prime divisors:

$$88 = 8 \times 11, \quad \gcd(8, 11) = 1$$

• Step 2: Apply divisibility by 8 to the last three digits ($8y6$): The three-digit number $8y6$ must be divisible by 8. Since $800$ is divisible by 8, the two-digit ending $y6$ must be divisible by 8. Multiples of 8 ending in 6 between 00 and 99 are: $16, 56, 96$. Therefore, $y \in \{1, 5, 9\}$.
• Step 3: Evaluate Candidate Values for $x$ under each $y$ via Rule for 11: Number: $3 4 2 x 1 8 y 6$. Sum of digits at odd places (from right): $S_1 = 6 + 8 + x + 4 = 18 + x$. Sum of digits at even places (from right): $S_2 = y + 1 + 2 + 3 = 6 + y$. Difference: $D = S_1 - S_2 = (18 + x) - (6 + y) = 12 + x - y$. For divisibility by 11, $12 + x - y$ must be a multiple of 11.
  • Case 1: $y = 9$ (maximizes $y$):

$$12 + x - 9 = 3 + x$$

For $3 + x$ to be a multiple of 11, $3 + x = 11 \implies x = 8$.
Calculate: $3x + 2y = 3(8) + 2(9) = 24 + 18 = 42$.

• Case 2: $y = 5$:

$$12 + x - 5 = 7 + x$$

For $7 + x = 11 \implies x = 4$.
Calculate: $3x + 2y = 3(4) + 2(5) = 12 + 10 = 22$.

• Case 3: $y = 1$:

$$12 + x - 1 = 11 + x$$

For $11 + x$ to be a multiple of 11, $x = 0$ (since $x < 10$).
Calculate: $3x + 2y = 3(0) + 2(1) = 2$.

• Conclusion: The maximum possible value of $(3x + 2y)$ is 42 (with $x = 8, y = 9$).

---

Worked Example 2: Complete Canonical Factor Analysis of $N = 3600$

Problem: For the integer $N = 3600$, determine: (a) Total number of factors, (b) Number of even factors, (c) Number of odd factors, (d) Sum of all factors, and (e) Sum of even factors.

• Step 1: Canonical Prime Factorization:

$$3600 = 36 \times 100 = (2^2 \times 3^2) \times (2^2 \times 5^2) = 2^4 \times 3^2 \times 5^2$$

Here $p_1 = 2, a = 4$; $p_2 = 3, b = 2$; $p_3 = 5, c = 2$.

• Step 2: Total number of factors:

$$T(3600) = (4 + 1)(2 + 1)(2 + 1) = 5 \times 3 \times 3 = 45$$

• Step 3: Number of odd factors: Exclude prime factor 2 entirely (set exponent of 2 to 0):

$$T_{\text{odd}} = (2 + 1)(2 + 1) = 3 \times 3 = 9$$

• Step 4: Number of even factors:

$$T_{\text{even}} = T - T_{\text{odd}} = 45 - 9 = 36$$

(Verification: $a(b+1)(c+1) = 4 \times 3 \times 3 = 36$.)

• Step 5: Sum of all factors:

$$S(3600) = \left(\frac{2^5 - 1}{2 - 1}\right) \times \left(\frac{3^3 - 1}{3 - 1}\right) \times \left(\frac{5^3 - 1}{5 - 1}\right)$$ $$S(3600) = 31 \times \frac{26}{2} \times \frac{124}{4} = 31 \times 13 \times 31 = 403 \times 31 = 12,493$$

• Step 6: Sum of even factors: Sum of powers of 2 starting at $2^1$:

$$(2^1 + 2^2 + 2^3 + 2^4) = 2 + 4 + 8 + 16 = 30$$ $$S_{\text{even}} = 30 \times 13 \times 31 = 30 \times 403 = 12,090$$

(Check: $S_{\text{odd}} = 1 \times 13 \times 31 = 403$, and $12,090 + 403 = 12,493$.)

---

Worked Example 3: Modular Remainder Evaluation

Problem: Find the remainder when $19^{102} + 28!$ is divided by 29.

• Step 1: Evaluate $19^{102} \pmod{29}$ using Fermat's Little Theorem: The modulus 29 is a prime number, and $\gcd(19, 29) = 1$. By Fermat's Little Theorem: $19^{29 - 1} = 19^{28} \equiv 1 \pmod{29}$. Divide the exponent 102 by 28:

$$102 = 28 \times 3 + 18$$

Therefore:

$$19^{102} = (19^{28})^3 \times 19^{18} \equiv 1^3 \times 19^{18} = 19^{18} \pmod{29}$$

Notice $19 \equiv -10 \pmod{29}$, and $19^2 = 361 = 29 \times 12 + 13 \equiv 13 \pmod{29}$. Also $13^2 = 169 = 29 \times 5 + 24 \equiv -5 \pmod{29}$. Then $19^{18} = (19^2)^9 \equiv 13^9 = (13^2)^4 \times 13 \equiv (-5)^4 \times 13 = 625 \times 13 \pmod{29}$. Since $625 = 29 \times 21 + 16 \equiv 16 \pmod{29}$:

$$16 \times 13 = 208 = 29 \times 7 + 5 \equiv 5 \pmod{29}$$

So $19^{102} \equiv 5 \pmod{29}$.

• Step 2: Evaluate $28! \pmod{29}$ using Wilson's Theorem: Since 29 is prime, Wilson's Theorem directly dictates:

$$(29 - 1)! = 28! \equiv -1 \equiv 28 \pmod{29}$$

• Step 3: Combine remainders:

$$19^{102} + 28! \equiv 5 + (-1) = 4 \pmod{29}$$

• Conclusion: The remainder is 4.