HCF AND LCM

Brief Overview

graph TD; A[Numbers] --> B[Factors]; A --> C[Multiples]; B --> D[HCF: Highest Common Factor]; C --> E[LCM: Least Common Multiple];

Introduction

Highest Common Factor (HCF) and Least Common Multiple (LCM) are fundamental arithmetic concepts dealing with the divisibility of numbers. They are widely used in simplifying fractions, scheduling, and solving cyclic pattern problems.

1. Prime Factorization Method

Concept: Every composite number can be expressed as a unique product of prime numbers. This is known as the Fundamental Theorem of Arithmetic.
Method: Repeatedly divide a number by prime numbers (2, 3, 5, 7, etc.) until the quotient is 1. Write the number as a product of these primes, often using exponents.

Numerical Example:
Find the prime factorization of 60.
$60 \div 2 = 30$
$30 \div 2 = 15$
$15 \div 3 = 5$
$5 \div 5 = 1$
Prime factorization of $60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3^1 \times 5^1$.

MCQs - Prime Factorization

Which of the following numbers is a prime number?
a) 9
b) 15
c) 2
d) 21
Answer: c
The prime factorization of 24 is:
a) $4 \times 6$
b) $2 \times 12$
c) $2^3 \times 3$
d) $2^2 \times 6$
Answer: c
Express 100 as a product of prime factors.
a) $10^2$
b) $2^2 \times 5^2$
c) $4 \times 25$
d) $2 \times 50$
Answer: b
The Fundamental Theorem of Arithmetic applies to:
a) All integers
b) Negative numbers only
c) Composite numbers
d) Fractions
Answer: c
What is the prime factorization of 45?
a) $5 \times 9$
b) $3^2 \times 5$
c) $15 \times 3$
d) $3 \times 5^2$
Answer: b
Which of these is not a prime number?
a) 2
b) 7
c) 1
d) 11
Answer: c
The prime factorization of 90 is:
a) $2 \times 3^2 \times 5$
b) $2 \times 9 \times 5$
c) $3 \times 30$
d) $9 \times 10$
Answer: a
If a number has a prime factorization of $2^3 \times 5^2$, what is the number?
a) 100
b) 200
c) 400
d) 500
Answer: b
What is the sum of the distinct prime factors of 30?
a) 8
b) 10
c) 15
d) 30
Answer: b
The only even prime number is:
a) 0
b) 2
c) 4
d) 6
Answer: b

2. Highest Common Factor (HCF)

Definition: Also known as Greatest Common Divisor (GCD). It is the largest positive integer that divides two or more numbers without leaving a remainder.
Finding HCF via Prime Factorization: Write the prime factorization of each number. The HCF is the product of the lowest powers of common prime factors.

Numerical Example:
Find the HCF of 24 and 36.
$24 = 2^3 \times 3^1$
$36 = 2^2 \times 3^2$
Common primes are 2 and 3. Lowest power of 2 is $2^2$. Lowest power of 3 is $3^1$.
$\text{HCF} = 2^2 \times 3^1 = 4 \times 3 = 12$.

MCQs - Highest Common Factor (HCF)

Another name for Highest Common Factor (HCF) is:
a) Least Common Multiple
b) Greatest Common Divisor
c) Lowest Common Denominator
d) Prime Factor
Answer: b
The HCF of 8 and 12 is:
a) 2
b) 4
c) 8
d) 24
Answer: b
What is the HCF of two prime numbers?
a) Their product
b) Their sum
c) 1
d) 0
Answer: c (Prime numbers only share 1 as a common factor)
To find the HCF using prime factorization, you multiply the:
a) Highest powers of all prime factors
b) Lowest powers of common prime factors
c) Highest powers of common prime factors
d) All prime factors without exponents
Answer: b
The HCF of 15, 30, and 45 is:
a) 5
b) 10
c) 15
d) 90
Answer: c
What is the HCF of 18 and 27?
a) 3
b) 6
c) 9
d) 54
Answer: c
Two numbers are co-prime if their HCF is:
a) 0
b) 1
c) Their product
d) 2
Answer: b
Find the greatest number that will divide 45 and 60 exactly.
a) 5
b) 15
c) 30
d) 60
Answer: b
The HCF of any two consecutive even numbers is always:
a) 1
b) 2
c) 4
d) Their difference
Answer: b
If HCF of $a$ and $b$ is $x$, then $x$ must be a factor of:
a) $a+b$
b) $a-b$
c) $a \times b$
d) All of the above
Answer: d

3. Least Common Multiple (LCM)

Definition: The smallest positive integer that is divisible by two or more numbers (i.e., it is a multiple of all of them).
Finding LCM via Prime Factorization: Write the prime factorization. The LCM is the product of the highest powers of all prime factors present in the numbers.
Relationship: $\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b$.

Numerical Example:
Find the LCM of 12 and 15.
$12 = 2^2 \times 3^1$
$15 = 3^1 \times 5^1$
Prime factors present are 2, 3, and 5. Highest powers: $2^2, 3^1, 5^1$.
$\text{LCM} = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60$.

MCQs - Least Common Multiple (LCM)

The LCM of 4 and 6 is:
a) 2
b) 12
c) 24
d) 10
Answer: b
To find the LCM using prime factorization, you multiply the:
a) Highest powers of all prime factors
b) Lowest powers of common prime factors
c) Highest powers of common prime factors
d) Lowest powers of all prime factors
Answer: a
If the HCF of two numbers is 6 and their product is 216, what is their LCM?
a) 36
b) 1296
c) 12
d) 72
Answer: a ($\text{LCM} = 216 / 6 = 36$)
The LCM of two prime numbers $p$ and $q$ is:
a) 1
b) $p + q$
c) $p \times q$
d) $|p - q|$
Answer: c
You have two blinking lights. One blinks every 4 seconds, the other every 5 seconds. How many seconds until they blink together again?
a) 9
b) 10
c) 20
d) 40
Answer: c (This is an LCM problem: LCM of 4 and 5 is 20)
What is the LCM of 14 and 21?
a) 7
b) 21
c) 42
d) 84
Answer: c
Three bells toll at intervals of 10, 15, and 20 minutes respectively. If they toll together at 10:00 AM, at what time will they toll together next?
a) 10:30 AM
b) 10:45 AM
c) 11:00 AM
d) 11:15 AM
Answer: c
The LCM of 15, 25, and 30 is:
a) 150
b) 120
c) 75
d) 5
Answer: a
Find the least number which is exactly divisible by 12, 16, and 24.
a) 48
b) 72
c) 96
d) 144
Answer: a
If $A$ is a multiple of $B$, what is the LCM of $A$ and $B$?
a) $A$
b) $B$
c) $A \times B$
d) $A / B$
Answer: a

Fun Facts about HCF and LCM!

Cicada Life Cycles: Some species of cicadas emerge every 13 or 17 years. These are prime numbers! This minimizes the chance (LCM) that their emergence will sync up with cycles of predators, keeping them safe.
Euclid's Algorithm: Finding the HCF of two large numbers using prime factorization can be extremely slow. The Greek mathematician Euclid invented a much faster division-based algorithm around 300 BC, and it's still used by computers today!
Gears and LCM: If you have two interlocking gears with a different number of teeth, the LCM of their teeth counts tells you exactly how many teeth must pass before they return to their exact starting alignment.

Hcf And Lcm