Number System questions with answers and concepts

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405 practice questionsPage updated 11 Nov 2026Reviewed by AISEA Editorial DeskPractice questions

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The Number System is a fundamental topic in quantitative aptitude that deals with the properties and operations of numbers. It encompasses natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. Mastery of divisibility rules, factors, multiples, prime factorisation, and unit digit patterns is essential for solving competitive exam questions efficiently.

Key formulas and rules

10 rules

$1 + 2 + \dots + n = \frac{n ( n + 1 )}{2}$

$1 + 3 + \dots + (2 n - 1) = n^{2}$

$2 + 4 + \dots + 2 n = n (n + 1)$

$1^{2} + 2^{2} + \dots + n^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}$

$1^{3} + 2^{3} + \dots + n^{3} = [\frac{n ( n + 1 )}{2}]^{2}$

$N = p^{a} q^{b} r^{c} \Rightarrow d (N) = (a + 1) (b + 1) (c + 1)$

$\prod_{d ∣ N} d = N^{d (N) /2}$

$lcm (a, b) g cd (a, b) = ∣ ab ∣$

$N mod 9 = (sum of the digits of N) mod 9$

$last two digits of N = N mod 100$

Key concepts

Divisibility Rules

2: Last digit is even. 3: Sum of digits divisible by 3. 4: Last two digits divisible by 4. 5: Last digit is 0 or 5. 6: Divisible by both 2 and 3. 7: Double last digit, subtract from rest, check divisibility. 8: Last three digits divisible by 8. 9: Sum of digits divisible by 9. 10: Last digit is 0. 11: Alternating sum of digits divisible by 11. 12: Divisible by both 3 and 4.

Prime Factorisation

Breaking down a number into its prime components. Every integer greater than 1 is either prime or can be expressed as a product of primes, uniquely apart from their order. For example, $72 = 2^{3} \times 3^{2}$ . This is crucial for finding HCF, LCM, the number of factors, and the sum of factors.

Factors and Multiples

A factor divides a number with zero remainder. A multiple is the product of that number and an integer. If $N = p^{a} q^{b}$ , then the number of positive factors is $(a + 1) (b + 1)$ and their sum is $\frac{p ^{a + 1} - 1}{p - 1} \times \frac{q ^{b + 1} - 1}{q - 1}$ .

Unit Digit Patterns

Cyclicity determines the units digit of powers. Digits 0, 1, 5 and 6 stay constant; 4 and 9 have cycles of length 2; 2, 3, 7 and 8 have cycles of length 4. To find the units digit of $a^{b}$ , reduce $b$ modulo the cycle length. For example, $47 mod 4 = 3$ , so $3^{47}$ has the same units digit as $3^{3}$ , which is 7.

Remainder Theorems

Euler's theorem states that $a^{φ (n)} \equiv 1 (mod n)$ when $g cd (a, n) = 1$ . Fermat's little theorem gives $a^{p - 1} \equiv 1 (mod p)$ when $p$ is prime and $p ∤ a$ . The Chinese Remainder Theorem solves compatible systems of congruences with pairwise-coprime moduli.

Types of Numbers

Natural numbers are $1, 2, 3, \dots$ ; whole numbers add 0; integers extend in both directions. Rational numbers have the form $p / q$ with $q \neq = 0$ . Irrational numbers cannot be written in that form, for example $2$ and $π$ . A prime integer greater than 1 has exactly two positive factors; a composite integer greater than 1 has more than two.

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Worked examples

Example 1

Find the number of positive divisors of 360 and the sum of all its divisors.

Solution

Step 1: Write the prime factorisation of 360.

360 = 2^{3} \times 3^{2} \times 5^{1}

Step 2: Calculate the number of divisors $d (N)$ .

d (360) = (3 + 1) (2 + 1) (1 + 1) = 4 \times 3 \times 2 = 24

Step 3: Calculate the sum of divisors $σ (N)$ .

σ (360) = (\frac{2 ^{4} - 1}{2 - 1}) \times (\frac{3 ^{3} - 1}{3 - 1}) \times (\frac{5 ^{2} - 1}{5 - 1})

σ (360) = 15 \times \frac{26}{2} \times \frac{24}{4} = 15 \times 13 \times 6 = 1, 170

Answer: 24 divisors with a sum of 1,170.

Example 2

What is the units digit of the expression $7^{2023} + 8^{2024} + 9^{2025}$ ?

Solution

Step 1: Determine the units digit of $7^{2023}$ .
The powers of 7 have a 4-cycle of units digits $(7, 9, 3, 1)$ :

2023 \equiv 3 (mod 4) ⟹ 7^{2023} \equiv 7^{3} \equiv 343 \equiv 3 (mod 10)

Step 2: Determine the units digit of $8^{2024}$ .
The powers of 8 have a 4-cycle $(8, 4, 2, 6)$ :

2024 \equiv 0 (mod 4) ⟹ 8^{2024} \equiv 8^{4} \equiv 4096 \equiv 6 (mod 10)

Step 3: Determine the units digit of $9^{2025}$ .
The powers of 9 have a 2-cycle $(9, 1)$ :

2025 \equiv 1 (mod 2) ⟹ 9^{2025} \equiv 9^{1} \equiv 9 (mod 10)

Step 4: Sum the components.

Sum of units digits = 3 + 6 + 9 = 18 ⟹ 8

Answer: 8.

Example 3

Find the remainder when $2^{100}$ is divided by 5.

Solution

Step 1: Inspect residues modulo 5.

2^{1} \equiv 2 (mod 5)

2^{2} \equiv 4 (mod 5)

2^{3} \equiv 3 (mod 5)

2^{4} \equiv 16 \equiv 1 (mod 5)

Step 2: Express the exponent as a multiple of 4.

100 = 4 \times 25

2^{100} = (2^{4})^{25} \equiv 1^{25} \equiv 1 (mod 5)

Answer: 1.

Example 4

A positive integer leaves a remainder of 3 when divided by 5, and a remainder of 2 when divided by 7. What is the smallest such positive integer?

Solution

Step 1: Formulate congruences.

N \equiv 3 (mod 5)

N \equiv 2 (mod 7)

Step 2: Generate solutions for $N \equiv 3 (mod 5)$ .

N \in {3, 8, 13, 18, 23, 28, \dots}

Step 3: Test values modulo 7.
- $3 \equiv 3 \neq \equiv 2 (mod 7)$
- $8 \equiv 1 \neq \equiv 2 (mod 7)$
- $13 \equiv 6 \neq \equiv 2 (mod 7)$
- $18 \equiv 4 \neq \equiv 2 (mod 7)$
- $23 \equiv 2 (mod 7)$ (satisfies both!)

Answer: 23.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

What is the unit digit of $2^{47}$ ?

Answer: D. 8

Explanation

Unit digits of powers of 2 cycle: 2, 4, 8, 6
$47/4 = 11$ remainder 3
3rd position in cycle = 8

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

How many factors does 72 have?

Answer: C. 12

Explanation

$72 = 2^{3} \times 3^{2}$
Number of factors = (3+1)(2+1) = 4x3 = 12

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

Which of the following is divisible by 11?

123456
121121
987654
13579

Answer: B. 121121

Explanation

For 121121, the alternating digit sum is $(1 + 1 + 2) - (2 + 1 + 1) = 0$ . Therefore 121121 is divisible by 11. The other options fail the divisibility test.
Answer: 121121

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

Memorise divisibility rules up to 12 thoroughly - they save significant calculation time
For unit digit problems, remember cyclicity patterns: 2,3,7,8 cycle every 4; 4,9 cycle every 2; 0,1,5,6 remain constant
When finding number of factors, always start with prime factorisation - it's the foundation for multiple concepts
For remainder problems with large exponents, look for patterns or use Euler's theorem when applicable
In competitive exams, estimation and digit analysis often help eliminate wrong options faster than full calculation
Practice mental calculation of digital roots (sum of digits mod 9) for quick divisibility checks and error detection

Sources and review notes

Reviewed by AISEA Editorial Desk on 28 Aug 2026.

Formula notation and worked examples were checked for complete exponents, congruence symbols and modulus conditions.
The references below support the number-theory claims; exercise questions may carry separate paper-level provenance.

Power SumWolfram MathWorld · Checked 28 Aug 2026Supports: the sums of consecutive powers listed in the formula section
Divisor FunctionWolfram MathWorld · Checked 28 Aug 2026Supports: factor-count and factor-sum formulas from prime factorisation
Euler's Totient TheoremWolfram MathWorld · Checked 28 Aug 2026Supports: Euler's theorem and its coprimality condition
Fermat's Little TheoremWolfram MathWorld · Checked 28 Aug 2026Supports: the prime-modulus exponent rule

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Next topic: HCF and LCM

Number System questions with answers and concepts

405 practice questionsPage updated 11 Nov 2026Reviewed by AISEA Editorial DeskPractice questions

Practise this concept

Key formulas and rules

10 rules

$1 + 2 + \dots + n = \frac{n ( n + 1 )}{2}$

$1 + 3 + \dots + (2 n - 1) = n^{2}$

$2 + 4 + \dots + 2 n = n (n + 1)$

$1^{2} + 2^{2} + \dots + n^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}$

$1^{3} + 2^{3} + \dots + n^{3} = [\frac{n ( n + 1 )}{2}]^{2}$

$N = p^{a} q^{b} r^{c} \Rightarrow d (N) = (a + 1) (b + 1) (c + 1)$

$\prod_{d ∣ N} d = N^{d (N) /2}$

$lcm (a, b) g cd (a, b) = ∣ ab ∣$

$N mod 9 = (sum of the digits of N) mod 9$

$last two digits of N = N mod 100$

Key concepts

Divisibility Rules

Prime Factorisation

Factors and Multiples

Unit Digit Patterns

Remainder Theorems

Types of Numbers

Check the idea while it is fresh.Open a focused exercise without losing this article.Practise this concept

Worked examples

Example 1

Find the number of positive divisors of 360 and the sum of all its divisors.

Solution

Step 1: Write the prime factorisation of 360.

360 = 2^{3} \times 3^{2} \times 5^{1}

Step 2: Calculate the number of divisors $d (N)$ .

d (360) = (3 + 1) (2 + 1) (1 + 1) = 4 \times 3 \times 2 = 24

Step 3: Calculate the sum of divisors $σ (N)$ .

σ (360) = (\frac{2 ^{4} - 1}{2 - 1}) \times (\frac{3 ^{3} - 1}{3 - 1}) \times (\frac{5 ^{2} - 1}{5 - 1})

σ (360) = 15 \times \frac{26}{2} \times \frac{24}{4} = 15 \times 13 \times 6 = 1, 170

Answer: 24 divisors with a sum of 1,170.

Example 2

What is the units digit of the expression $7^{2023} + 8^{2024} + 9^{2025}$ ?

Solution

Step 1: Determine the units digit of $7^{2023}$ .
The powers of 7 have a 4-cycle of units digits $(7, 9, 3, 1)$ :

2023 \equiv 3 (mod 4) ⟹ 7^{2023} \equiv 7^{3} \equiv 343 \equiv 3 (mod 10)

Step 2: Determine the units digit of $8^{2024}$ .
The powers of 8 have a 4-cycle $(8, 4, 2, 6)$ :

2024 \equiv 0 (mod 4) ⟹ 8^{2024} \equiv 8^{4} \equiv 4096 \equiv 6 (mod 10)

Step 3: Determine the units digit of $9^{2025}$ .
The powers of 9 have a 2-cycle $(9, 1)$ :

2025 \equiv 1 (mod 2) ⟹ 9^{2025} \equiv 9^{1} \equiv 9 (mod 10)

Step 4: Sum the components.

Sum of units digits = 3 + 6 + 9 = 18 ⟹ 8

Answer: 8.

Example 3

Find the remainder when $2^{100}$ is divided by 5.

Solution

Step 1: Inspect residues modulo 5.

2^{1} \equiv 2 (mod 5)

2^{2} \equiv 4 (mod 5)

2^{3} \equiv 3 (mod 5)

2^{4} \equiv 16 \equiv 1 (mod 5)

Step 2: Express the exponent as a multiple of 4.

100 = 4 \times 25

2^{100} = (2^{4})^{25} \equiv 1^{25} \equiv 1 (mod 5)

Answer: 1.

Example 4

A positive integer leaves a remainder of 3 when divided by 5, and a remainder of 2 when divided by 7. What is the smallest such positive integer?

Solution

Step 1: Formulate congruences.

N \equiv 3 (mod 5)

N \equiv 2 (mod 7)

Step 2: Generate solutions for $N \equiv 3 (mod 5)$ .

N \in {3, 8, 13, 18, 23, 28, \dots}

Answer: 23.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

What is the unit digit of $2^{47}$ ?

Answer: D. 8

Explanation

Unit digits of powers of 2 cycle: 2, 4, 8, 6
$47/4 = 11$ remainder 3
3rd position in cycle = 8

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

How many factors does 72 have?

Answer: C. 12

Explanation

$72 = 2^{3} \times 3^{2}$
Number of factors = (3+1)(2+1) = 4x3 = 12

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

Which of the following is divisible by 11?

123456
121121
987654
13579

Answer: B. 121121

Explanation

For 121121, the alternating digit sum is $(1 + 1 + 2) - (2 + 1 + 1) = 0$ . Therefore 121121 is divisible by 11. The other options fail the divisibility test.
Answer: 121121

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

Memorise divisibility rules up to 12 thoroughly - they save significant calculation time
For unit digit problems, remember cyclicity patterns: 2,3,7,8 cycle every 4; 4,9 cycle every 2; 0,1,5,6 remain constant
When finding number of factors, always start with prime factorisation - it's the foundation for multiple concepts
For remainder problems with large exponents, look for patterns or use Euler's theorem when applicable
In competitive exams, estimation and digit analysis often help eliminate wrong options faster than full calculation
Practice mental calculation of digital roots (sum of digits mod 9) for quick divisibility checks and error detection

Sources and review notes

Reviewed by AISEA Editorial Desk on 28 Aug 2026.

Formula notation and worked examples were checked for complete exponents, congruence symbols and modulus conditions.
The references below support the number-theory claims; exercise questions may carry separate paper-level provenance.

Power SumWolfram MathWorld · Checked 28 Aug 2026Supports: the sums of consecutive powers listed in the formula section
Divisor FunctionWolfram MathWorld · Checked 28 Aug 2026Supports: factor-count and factor-sum formulas from prime factorisation
Euler's Totient TheoremWolfram MathWorld · Checked 28 Aug 2026Supports: Euler's theorem and its coprimality condition
Fermat's Little TheoremWolfram MathWorld · Checked 28 Aug 2026Supports: the prime-modulus exponent rule

Ready to test your understanding?

Work through 405 questions with explanations after each answer.

Start practice

Next topic: HCF and LCM

Power SumWolfram MathWorld · Checked 28 Aug 2026Supports: the sums of consecutive powers listed in the formula section

Divisor FunctionWolfram MathWorld · Checked 28 Aug 2026Supports: factor-count and factor-sum formulas from prime factorisation

Euler's Totient TheoremWolfram MathWorld · Checked 28 Aug 2026Supports: Euler's theorem and its coprimality condition

Fermat's Little TheoremWolfram MathWorld · Checked 28 Aug 2026Supports: the prime-modulus exponent rule

Key formulas and rules

Key concepts

Divisibility Rules

Prime Factorisation

Factors and Multiples

Unit Digit Patterns

Remainder Theorems

Types of Numbers

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Sources and review notes

Ready to test your understanding?

Keep building the same skill family.

Key formulas and rules

Key concepts

Divisibility Rules

Prime Factorisation

Factors and Multiples

Unit Digit Patterns

Remainder Theorems

Types of Numbers

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Sources and review notes

Ready to test your understanding?

Keep building the same skill family.