Algebra

Number system

Q: FAQ 1: Why is $1$ not considered a prime number in the Fundamental Theorem of Arithmetic?

If $1$ were prime, prime factorizations would no longer be unique, as any number could be multiplied by $1$ an arbitrary number of times (e.g., $6 = 2 \times 3 = 1 \times 2 \times 3 = 1^2 \times 2 \times 3$). Uniqueness is preserved strictly by defining primes as integers greater than 1 with exactly two distinct positive divisors.

Q: FAQ 2: What is the most common reason students lose marks on number system questions?

Calculation slips during prime factorization of large numbers and confusing total divisors with proper or even/odd divisors. Candidates also frequently fail to check whether denominators are fully reduced before applying totient or decimal termination rules.

Q: FAQ 3: How can candidates solve these questions faster during the exam?

Memorize standard exponent cyclicity patterns (especially powers of 2, 3, 7, 9 modulo 10), practice mental prime factorization up to 1000, and employ digital roots (modulo 9) to verify or eliminate options rapidly.

The number system forms the foundational bedrock of algebraic structures, branching progressively from the set of natural numbers $\mathbb{N}$ to integers $\mathbb{Z}$, rational numbers $\mathbb{Q}$, real numbers $\mathbb{R}$, and complex numbers $\mathbb{C}$. This topic covers divisibility criteria, prime factorization, modular arithmetic, and structural field axioms.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

Fundamental Theorem of Arithmetic (Unique Factorization Theorem)

Every integer $n > 1$ can be uniquely represented as a product of prime numbers, up to the order of the prime factors: $$n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k} = \prod_{i=1}^{k} p_i^{a_i}$$ where $p_1 < p_2 < \dots < p_k$ are distinct primes and $a_i \in \mathbb{N}$ are positive integers.

📐 Geometric Interpretation

Geometrically, any rectangular grid configuration of composite area $n$ can be broken down uniquely into prime-dimensional irreducible prime blocks, reflecting unique factorization along discrete lattices.

⚙️ Algebraic & Structural Meaning

Establishes the Euclidean domain properties of the ring of integers $\mathbb{Z}$, guaranteeing that greatest common divisors (GCD) and least common multiples (LCM) are structurally linked via $\text{gcd}(a,b) \cdot \text{lcm}(a,b) = |ab|$.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Integer Domain Requirement ($n \in \mathbb{Z}, n > 1$)	`Discrete set ${2, 3, 4, \dots}$`	Strictly Mandatory / Necessary	Units ($1, -1$) and zero ($0$) or fractions do not possess prime factorizations, invalidating the unique product structure.
Primality of factors ($p_i$ is prime)	`Prime subset of $\mathbb{N}$`	Strictly Mandatory	If composite factors are permitted, factorization is no longer unique (e.g., $6 = 2 \times 3 = (-2) \times (-3)$ or non-unique in broader rings like $\mathbb{Z}[\sqrt{-5}]$).
Positivity of exponents ($a_i \ge 1$)	`Positive integers $\mathbb{Z}^+$`	Essential for canonical representation	Without canonical ordering and positive exponents, representations become infinite or ambiguous.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Total Number of Divisors

$$ \text{Total Divisors} = \prod_{i=1}^{k} (a_i + 1) $$

Rule: Add 1 to each exponent in the canonical prime factorization and find their product.

Application: Directly tested in UP PGT/TGT to find divisor counts of large numbers.

Sum of Divisors

$$ \sigma(n) = \prod_{i=1}^{k} \frac{p_i^{a_i + 1} - 1}{p_i - 1} $$

Rule: Compute the geometric series sum for each prime power factor independently and multiply them.

Number of Coprime Divisors / Euler's Totient Count

$$ \phi(n) = n \prod_{p|n} \left(1 - \frac{1}{p}\right) $$

Rule: Multiplicative formula to find positive integers $\le n$ that are relatively prime to $n$.

Application: Standard question type in UP TGT/PGT algebraic number theory sections.

Trailing Zeros in $n!$

$$ E_5(n!) = \sum_{k=1}^{\infty} \lfloor \frac{n}{5^k} \rfloor = \lfloor \frac{n}{5} \rfloor + \lfloor \frac{n}{25} \rfloor + \lfloor \frac{n}{125} \rfloor + \dots $$

Rule: Repeatedly integer-divide $n$ by 5 until the quotient becomes zero, and sum the quotients.

Unit Digit of Powers ($a^b \pmod{10}$)

$$ a^b \pmod{10} \equiv a^{b \pmod{\phi(10)}} \pmod{10} \equiv a^{b \pmod{4}} \pmod{10} $$

Rule: Cyclicity of unit digits for integers repeats every 4 powers (except for numbers ending in 0, 1, 5, 6). Divide exponent by 4.

Rapid Elimination via Modulo 9 (Digital Root)

$$ x \equiv \text{sum of digits of } x \pmod{9} $$

Rule: Arithmetic operations on numbers preserve congruence modulo 9. Use digital roots to eliminate wrong options instantly.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Prime vs Composite Misclassification of 1

Common Pitfall: Candidates frequently treat $1$ as a prime number or include it in prime factorizations.

Counterexample: $1$ is neither prime nor composite.

Why it fails: By definition, a prime number must have exactly two distinct positive divisors. 1 has only one divisor.

🛡️ Exam Defense: Never include 1 in prime factorization or prime counting formulas.

⚠️ COMMON ERROR #02

Misapplying Euler's Totient to Non-Canonical Forms

Common Pitfall: Applying the formula $\phi(n)$ without first completely reducing $n$ into distinct prime powers.

Counterexample: Calculating $\phi(10)$ directly as $10(1 - 1/10)$ vs correct factor breakdown $10 = 2^1 \cdot 5^1$.

Why it fails: Product runs over distinct prime divisors $p|n$, not composite divisors.

🛡️ Exam Defense: Always prime-factorize $n$ completely before applying $\phi(n)$ or divisor sum formulas.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Natural & Integers ($\mathbb{N}, \mathbb{Z}$)	Closed under addition & multiplication	Divisibility: $a = bq + r$	Discrete lattice points on a 1D line	Divisibility, remainders, HCF/LCM
Rational Numbers ($\mathbb{Q}$)	Field axioms, terminating/recurring decimals	$p/q, q \neq 0$	Dense countable set on real line	Decimal expansion conversion
Real Numbers ($\mathbb{R}$)	Ordered field with Least Upper Bound property	Completeness axiom	Continuous unbroken continuum line	Supremum, infimum, bounds
Complex Numbers ($\mathbb{C}$)	Algebraically closed field	$z = x + iy, i^2 = -1$	2D Argand plane coordinates	De Moivre's theorem, roots of unity

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Find the total number of even divisors of the number $1080$.

(A) 24(B) 20(C) 16(D) 32

📚 Method A: Formal Textbook Derivation

Step 1: Prime factorize $1080 = 108 \times 10 = (2^2 \cdot 3^3) \cdot (2^1 \cdot 5^1) = 2^3 \cdot 3^3 \cdot 5^1$. Step 2: For an even divisor, factor 2 out: $2(2^2 \cdot 3^3 \cdot 5^1)$. Step 3: Total even divisors = $(2+1)(3+1)(1+1) = 3 \times 4 \times 2 = 24$? Wait, let's recalculate: $1080 = 2^3 \cdot 3^3 \cdot 5^1$. Even divisors must have at least one factor of 2, so choose $2^k$ where $k \in \{1, 2, 3\}$ (3 ways), $3^m$ ($m \in \{0, 1, 2, 3\}$, 4 ways), $5^p$ ($p \in \{0, 1\}$, 2 ways). Total even divisors = $3 \times 4 \times 2 = 24$. Total overall divisors = $(3+1)(3+1)(1+1) = 32$. Odd divisors = $32 - 24 = 8$. Correct option is (A).

⚡ Method B: 25-Second Exam Shortcut

Total divisors = $4 \times 4 \times 2 = 32$. Odd divisors (ignoring factor of 2) = $4 \times 2 = 8$. Even divisors = Total - Odd = $32 - 8 = 24$.

💡 Tip: Always compute total and subtract odd divisors to avoid combinatorial counting errors for even divisors.

CANONICAL PROBLEM 02 Dual-Method Breakdown

What is the remainder when $3^{2024}$ is divided by $5$?

(A) 1(B) 2(C) 3(D) 4

📚 Method A: Formal Textbook Derivation

Step 1: Use Fermat's Little Theorem or observe cyclicity of powers of 3 modulo 5. $3^1 \equiv 3$, $3^2 \equiv 4$, $3^3 \equiv 2$, $3^4 \equiv 1 \pmod 5$. Step 2: Cycle length is 4. Step 3: Divide exponent 2024 by 4: $2024 = 4 \times 506$, remainder is 0. Step 4: $3^{2024} \equiv (3^4)^{506} \equiv 1^{506} \equiv 1 \pmod 5$.

⚡ Method B: 25-Second Exam Shortcut

Check exponent mod 4: $2024 \pmod 4 = 0$, so $3^4 = 81 \equiv 1 \pmod 5$. Instant tick on (A).

💡 Tip: For prime modulus $p$, $a^{p-1} \equiv 1 \pmod p$ by Fermat's theorem when $\text{gcd}(a,p)=1$.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: The number of trailing zeros in $100!$ is:

A 20

B 24

C 25

D 30

Derivation & Analysis:

$E_5(100!) = \lfloor 100/5 \rfloor + \lfloor 100/25 \rfloor = 20 + 4 = 24$. Correct option is (B).

MCQ 2: If $\phi(n)$ denotes Euler's totient function and $\phi(p^k)$ for prime $p$ is:

A $p^k - p^{k-1}$

B $p^k - 1$

C $p^{k-1}(p-1)$

D Both (A) and (C) are correct

Derivation & Analysis:

Both $p^k - p^{k-1}$ and $p^{k-1}(p-1)$ are algebraically identical forms for $\phi(p^k)$. Correct option is (D).

MCQ 3: The unit digit of the expression $7^{95} - 3^{58}$ is:

A 4

B 6

C 2

D 8

Derivation & Analysis:

For $7^{95}$: $95 \pmod 4 = 3$, so unit digit of $7^3$ is $3$. For $3^{58}$: $58 \pmod 4 = 2$, so unit digit of $3^2$ is $9$. Difference = $3 - 9 = -6$. Adding 10 gives $10 - 6 = 4$? Wait, $3 - 9 = -6$, $13 - 9 = 4$. Let's re-verify: $7^1=7, 7^2=9, 7^3=3, 7^4=1$. $95 = 4(23)+3 \implies 3$. $3^1=3, 3^2=9, 3^3=7, 3^4=1$. $58 = 4(14)+2 \implies 9$. Thus $3 - 9 = -6 \equiv 4 \pmod{10}$. Correct option is (A).

MCQ 4: Which of the following sets is uncountable?

A The set of rational numbers $\mathbb{Q}$

B The set of algebraic numbers

C The set of real numbers $\mathbb{R}$

D The set of integers $\mathbb{Z}$

Derivation & Analysis:

By Cantor's diagonal argument, $\mathbb{R}$ is uncountable, whereas $\mathbb{Q}$, $\mathbb{Z}$, and algebraic numbers are countable. Correct option is (C).

MCQ 5: The HCF and LCM of two numbers are 12 and 72 respectively. If one number is 24, the other number is:

A 36

B 48

C 60

D 72

Derivation & Analysis:

Using $\text{HCF} \times \text{LCM} = a \times b$: $12 \times 72 = 24 \times b \implies b = \frac{12 \times 72}{24} = 36$. Correct option is (A).

MCQ 6: The greatest integer less than or equal to $x$ is denoted by $\lfloor x \rfloor$. The value of $\lfloor -3.4 \rfloor$ is:

A -3

B -4

C -3.4

D 3

Derivation & Analysis:

By definition of floor function, $\lfloor -3.4 \rfloor = -4$ because $-4 \le -3.4 < -3$. Correct option is (B).

MCQ 7: If $p$ is a prime number and $p \mid ab$, then:

A $p \mid a$ only

B $p \mid b$ only

C $p \mid a$ or $p \mid b$

D $p \mid a$ and $p \mid b$

Derivation & Analysis:

This is Euclid's Lemma for primes: if a prime divides a product, it must divide at least one of the factors. Correct option is (C).

MCQ 8: The decimal expansion of a rational number $\frac{p}{q}$ (where $\text{gcd}(p,q)=1$) terminates if and only if the prime factorization of $q$ is of the form:

A $2^n 3^m$

B $2^n 5^m$

C $3^n 5^m$

D $p^n q^m$

Derivation & Analysis:

A rational number has a terminating decimal expansion if and only if the prime factors of the denominator are strictly 2 and/or 5. Correct option is (B).

MCQ 9: The sum of all divisors of $360$ is:

A 1170

B 1080

C 960

D 1116

Derivation & Analysis:

$360 = 2^3 \cdot 3^2 \cdot 5^1$. Sum = $\frac{2^4-1}{2-1} \cdot \frac{3^3-1}{3-1} \cdot \frac{5^2-1}{5-1} = 15 \cdot 13 \cdot 6 = 1170$. Correct option is (A).

Q10

MCQ 10: Which of the following is a transcendental number?

A $\sqrt{2}$

B $\pi$

C $\frac{22}{7}$

D $\sqrt[3]{5}$

Derivation & Analysis:

$\pi$ is transcendental (proven by Lindemann), whereas $\sqrt{2}$, $\sqrt[3]{5}$ are algebraic irrational numbers and $22/7$ is rational. Correct option is (B).

Common questions

If $1$ were prime, prime factorizations would no longer be unique, as any number could be multiplied by $1$ an arbitrary number of times (e.g., $6 = 2 \times 3 = 1 \times 2 \times 3 = 1^2 \times 2 \times 3$). Uniqueness is preserved strictly by defining primes as integers greater than 1 with exactly two distinct positive divisors.

Calculation slips during prime factorization of large numbers and confusing total divisors with proper or even/odd divisors. Candidates also frequently fail to check whether denominators are fully reduced before applying totient or decimal termination rules.

Memorize standard exponent cyclicity patterns (especially powers of 2, 3, 7, 9 modulo 10), practice mental prime factorization up to 1000, and employ digital roots (modulo 9) to verify or eliminate options rapidly.

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