Relationships between Numbers
What is Relationships between Numbers?
A number that divides another number exactly, without leaving a remainder.
Key formula / rule: HCF and LCM Relationship
Key points
- To master divisibility rules for quick calculations.
- To understand and apply HCF and LCM concepts to solve problems.
- To perform prime factorization accurately.
- To identify and utilize properties of various number types.
Common exam trap
Confusing HCF and LCM in problem statements.
Definitions
- Term
Factor
- Meaning
A number that divides another number exactly, without leaving a remainder.
- Term
Multiple
- Meaning
A number that can be divided by another number exactly. It is the product of the given number and an integer.
- Term
Prime Number
- Meaning
A natural number greater than 1 that has no positive divisors other than 1 and itself.
- Term
Composite Number
- Meaning
A natural number greater than 1 that is not prime; it has at least one divisor other than 1 and itself.
- Term
HCF (Highest Common Factor)
- Meaning
The largest positive integer that divides two or more integers without leaving a remainder.
- Term
LCM (Lowest Common Multiple)
- Meaning
The smallest positive integer that is a multiple of two or more integers.
- Term
Prime Factorization
- Meaning
The process of finding the prime numbers that multiply together to make the original number.
Learning objectives
To master divisibility rules for quick calculations.
To understand and apply HCF and LCM concepts to solve problems.
To perform prime factorization accurately.
To identify and utilize properties of various number types.
To solve problems involving number relationships efficiently.
Formulae
- Name
HCF and LCM Relationship
- Note
Applies to two numbers. For three numbers, HCF(a,b,c) × LCM(a,b,c) is not necessarily equal to a × b × c.
- Expression
HCF(a, b) × LCM(a, b) = a × b
- Name
Prime Factorization Method for HCF
- Note
Example: 12 = 22 * 3, 18 = 2 * 32. HCF = 21 * 31 = 6.
- Expression
Find prime factors of each number. HCF is the product of the lowest powers of common prime factors.
- Name
Prime Factorization Method for LCM
- Note
Example: 12 = 22 * 3, 18 = 2 * 32. LCM = 22 * 32 = 36.
- Expression
Find prime factors of each number. LCM is the product of the highest powers of all prime factors.
Prerequisites
Basic Arithmetic Operations (Addition, Subtraction, Multiplication, Division)
Understanding of Integers
Concept of Prime and Composite Numbers
Common mistakes
Confusing HCF and LCM in problem statements.
Incorrect application of divisibility rules, especially for 7, 11, and 13.
Assuming properties of numbers that are not universally true (e.g., sum of primes is prime).
Calculation errors in prime factorization or HCF/LCM.
Forgetting the relationship HCF x LCM = Product of numbers.
Keywords
Divisibility
Factors
Multiples
HCF
LCM
Prime Numbers
Composite Numbers
Number Theory
Arithmetic
Integers
Practice preview
A number is divisible by 3 if the sum of its digits is divisible by 3. Which of the following numbers is divisible by 3?…
easy
What is the Highest Common Factor (HCF) of 24 and 36?…
easy
Which of the following is a prime number?…
easy