Number System

Divisibility, factors, prime numbers, unit digit

625 practice questionsPage updated 27 Aug 2026Reviewed by AISEA Editorial DeskPractice questions

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

1+2+\cdots+n=\frac{n(n+1)}{2}

1+3+\cdots+(2n-1)=n^2

2+4+\cdots+2n=n(n+1)

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

1^3+2^3+\cdots+n^3=\left[\frac{n(n+1)}{2}\right]^2

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

\prod_{d\mid N} d=N^{d(N)/2}

\operatorname{lcm}(a,b)\,\gcd(a,b)=|ab|

N\bmod 9=(\text{sum of the digits of }N)\bmod 9

\text{last two digits of }N=N\bmod 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\times3^2

. This is crucial for finding HCF, LCM, the number of factors, and the sum of factors.

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

Number System

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

Example 2

Example 3

Example 4

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Sources and review notes

Ready to test your understanding?

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

Example 2

Example 3

Example 4

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Sources and review notes

Ready to test your understanding?

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