Number System: Concepts, Formulas, Tricks and Questions

Number System covers natural, whole, integer, rational, irrational and real numbers, along with divisibility, factors, multiples, HCF, LCM, remainders and unit digits. This page presents number system basics through definitions, number system formulas, classification rules, shortcuts and solved numerical examples used in quantitative aptitude questions.

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What is the Number System?

The Number System is a classification of numbers based on their properties and representations. It includes natural, whole, integer, rational, irrational and real numbers.

Natural numbers are counting numbers: 1, 2, 3, and so on. Whole numbers include zero with the natural numbers. Integers include positive numbers, negative numbers and zero. Rational numbers can be written as p/q, where p and q are integers and q is not zero. Irrational numbers cannot be expressed in this form. The set of real numbers contains both rational and irrational numbers.

Number System Formula & Tricks

Important Formulas

Euclidean Division Formula
a = bq + r, where 0 ≤ r < b

When a positive integer a is divided by a positive integer b, q is the quotient and r is the remainder.

HCF and LCM Relation
HCF(a, b) × LCM(a, b) = a × b

This relation applies to two positive integers a and b.

Sum of First n Natural Numbers
1 + 2 + 3 + ... + n = n(n + 1)/2

Use this formula to find the sum of the first n positive integers.

Sum of First n Odd Numbers
1 + 3 + 5 + ... + (2n − 1) = n²

The sum of the first n odd natural numbers is the square of n.

Sum of First n Even Numbers
2 + 4 + 6 + ... + 2n = n(n + 1)

The sum of the first n positive even numbers is n multiplied by n + 1.

Remainder of a Sum or Product
(a + b) mod m = [(a mod m) + (b mod m)] mod m

Remainders can be reduced before adding or multiplying numbers modulo m.

Quick Tricks

Use the divisibility rules before long division

Check simple rules first. A number is divisible by 3 or 9 when the sum of its digits is divisible by 3 or 9. It is divisible by 11 when the difference between the sums of alternate digits is 0 or a multiple of 11.

Example: The digit sum of 7,236 is 18, so 7,236 is divisible by both 3 and 9.
Find the unit digit through cycles

The unit digits of powers repeat in cycles. For example, powers of 2 have the cycle 2, 4, 8, 6. Divide the exponent by 4 and use the remainder to select the cycle position.

Example: The unit digit of 2^13 is the same as the unit digit at position 1 in the cycle, so it is 2.
Use the complement for powers of numbers ending in 5

For a number of the form 10a + 5, its square is a(a + 1) followed by 25.

Example: 35² = 3 × 4 followed by 25 = 1225.
Apply the remainder pattern for powers

If a number leaves a small remainder when divided by a divisor, replace the number with that remainder before calculating powers, while retaining the same modulus.

Example: Since 17 leaves remainder 2 when divided by 5, 17² leaves the same remainder as 2², which is 4.

Number System Concepts

Classification of Numbers

Natural numbers begin with 1, whole numbers include 0, and integers include negative numbers as well as positive numbers and zero.

Natural numbers: {1, 2, 3, ...}. Whole numbers: {0, 1, 2, 3, ...}. Integers: {..., −3, −2, −1, 0, 1, 2, 3, ...}. Rational numbers have the form p/q with q ≠ 0. Irrational numbers include √2 and π. Rational and irrational numbers together form the real numbers.

Example: −4 is an integer and a rational number because it can be written as −4/1. √3 is irrational because it cannot be written as p/q.

Factors, Multiples and Prime Numbers

A factor divides a number exactly, while a multiple is obtained by multiplying a number by an integer.

The factors of 12 are 1, 2, 3, 4, 6 and 12. Multiples of 12 are 12, 24, 36, and so on. A prime number has exactly two positive factors, 1 and itself. The number 1 is neither prime nor composite. Every integer greater than 1 has a unique prime factorisation apart from the order of its factors.

Example: 60 = 2² × 3 × 5. Its prime factors are 2, 3 and 5.

Divisibility Rules

Divisibility rules determine whether a number divides exactly without performing complete division.

A number is divisible by 2 if its last digit is even, by 5 if its last digit is 0 or 5, and by 10 if its last digit is 0. It is divisible by 4 if its last two digits form a number divisible by 4, and by 8 if its last three digits form a number divisible by 8. Divisibility by 6 requires divisibility by both 2 and 3.

Example: 5,724 is divisible by 4 because 24 is divisible by 4. It is also divisible by 6 because it is even and its digit sum, 18, is divisible by 3.

HCF and LCM

The HCF is the greatest common factor of given numbers, while the LCM is the least positive common multiple.

Using prime factorisation, the HCF is found by taking the smallest powers of common prime factors. The LCM is found by taking the greatest powers of all prime factors. For two positive integers, HCF × LCM equals their product.

Example: For 18 = 2 × 3² and 24 = 2³ × 3, HCF = 2 × 3 = 6 and LCM = 2³ × 3² = 72.

Remainders and Unit Digits

In division, the remainder is always non-negative and smaller than the divisor; unit-digit questions are solved using repeating last-digit cycles.

If a is divided by b, then a = bq + r, where 0 ≤ r < b. For powers, reduce the exponent according to the cycle length of the unit digits. The cycle length may be 1, 2 or 4 for many commonly used bases.

Example: The unit digits of powers of 3 repeat as 3, 9, 7, 1. Therefore, 3^10 has the same unit digit as the second term of the cycle, which is 9.

Number System Video Lessons

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Prime Numbers: Test and Find Them Fast

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Practice Number System Questions

Practise published questions related to this topic.

1The number 657792025 is divisible by which of the following numbers?→ 2What is the smallest number which when increased by 29 is exactly divisible by 18, 12, 15 and 20?→ 395P9 is divisible by 11 . What is the value of P ?→ 4The 7-digit number 77A444B is divisible by 24. What is the maximum value of (A + B)?→ 5The LCM of the numbers 7, 12.4 and 1 is:→ 6If x = √196 ÷ √49 + √81, then what is the value of 2x?→ 7A school drum beats every 12 minutes, and the school bell rings every 18 minutes. If both sounded together at 9:00 AM, how many times will they sound together again before 11:00 AM?→ 8What is the value of √(5 + 2√6) ?→ 9Find the HCF of 148, 172, and 198.→ 10What is the Least Common Multiple of all prime numbers between 5 and 12?→

Number System Quick Quiz

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Quick Revision Notes

Number System Revision Points

Keep these definitions, formulas and rules ready for quick calculation.

  • Natural numbers start from 1; whole numbers start from 0; integers include negative numbers, zero and positive numbers.
  • A rational number has the form p/q, where p and q are integers and q ≠ 0. Irrational numbers cannot be represented in this form.
  • A prime number has exactly two positive factors. The number 1 is neither prime nor composite.
  • For division, a = bq + r with 0 ≤ r < b.
  • HCF uses the smallest powers of common prime factors; LCM uses the greatest powers of all prime factors.
  • For two positive integers, HCF × LCM = product of the numbers.
  • A number is divisible by 3 or 9 when its digit sum is divisible by 3 or 9.
  • For unit digits of powers, identify the repeating cycle and reduce the exponent accordingly.

Number System FAQs

Is 0 a natural number?

Under the common competitive-examination convention, natural numbers are 1, 2, 3, ... and do not include 0. Whole numbers include 0.

What is the condition for a number to be divisible by 6?

The number must be divisible by both 2 and 3. Therefore, it must be even and its digit sum must be divisible by 3.

How is the HCF of 36 and 48 calculated?

36 = 2² × 3 and 48 = 2⁴ × 3. Taking the smallest powers gives HCF = 2² × 3 = 12.

What is the LCM of 12 and 18?

12 = 2² × 3 and 18 = 2 × 3². Taking the greatest powers gives LCM = 2² × 3² = 36.

What is the remainder when 157 is divided by 12?

157 = 12 × 13 + 1, so the remainder is 1.

What is the unit digit of 7^25?

The unit-digit cycle of 7 is 7, 9, 3, 1. Since 25 leaves remainder 1 when divided by 4, the unit digit is 7.

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