The document discusses the greatest common divisor (GCD), which is the largest positive integer that divides two or more numbers. There are several methods to find the GCD, including prime factorization, long division, and Euclid's algorithm. Prime factorization involves writing each number as a product of prime numbers and finding the largest common factor. Long division repeatedly divides the largest number by the next largest until the remainder is zero. Examples are provided to demonstrate finding the GCD using these methods. The GCD has applications in problems involving arranging items in equal groups or sections.
2. Introduction
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The greatest common divisor
(GCD) of two or more numbers is the
greatest common factor number that
divides them, exactly. It is also called
the highest common factor (HCF). For
example, the greatest common factor of 15
and 10 is 5, since both the numbers can be
divided by 5.
• 15/5 = 3
• 10/5 = 2
If a and b are two numbers then
the greatest common divisor of both the
numbers is denoted by gcd(a, b). To find
the gcd of numbers, we need to list all the
factors of the numbers and find the largest
common factor.
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Methods to Find
GCD
There are several methods to find
the greatest common divisor of
given two numbers.
1.Prime factorisation method
2.Long division method
3.Euclid’s division algorithm
4. Prime factorisation method
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• Every composite number, i.e. a number with more than one factor can be written as a product of
prime numbers.
• In the prime factorisation method, each given number is written as the product of prime
numbers and then find the product of the smallest power of each common prime factor.
• This method is applicable only for positive numbers, i,e. Natural numbers.
Example: Find the Greatest common factor of 24, 30 and 36.
Solution: Prime factors of 24 is 23 × 3
Prime factors of 30 = 2 × 3 × 5
Prime factors of 36 = 2² x 3²
From the factorisation, we can see, only 2 x 3 are common prime factors.
Therefore, GCD (24, 30, 36) = 2 x 3 = 6
5. Long division method
In this method, the largest number among the given set of
numbers should be divided by the second largest number, and
again the second-largest number should be divided by the
remainder of the previous operation, this process will continue till
the remainder is zero. The divisor, when the remainder is zero, is
called the greatest common divisor of the given numbers.
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6. Formula
If a and b are any number, then the greatest common divisor of a and b
can be given by:
𝐺𝐶𝐷 𝑎, 𝑏 =
[|𝑎. 𝑏|]
[𝑙𝑐𝑚(𝑎, 𝑏)]
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7. Application
• The concept of the greatest common divisor or the highest common factor
is used in many real-life incidents as below.
• A shopkeeper has 420 balls and 130 bats to pack in a day. She wants to
pack them in such a way that each set has the same number in a box, and
they take up the least area of the box. What is the number that can be
placed in each set for this packing purpose?
• In the above problem, the greatest common divisor of 420 and 130 will be
the required number.
• Other application like arranging students in rows and columns in equal
number, diving a group of people into smaller sections,etc.,
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8. Example 1: Find the greatest common divisor (or
HCF) of 128 and 96.
By the method of prime factorisation,
• 128 = 2 x 2 x 2 x 2 x 2 x 2 x 2
• 96 = 2 x 2 x 2 x 2 x 2 x 3
• HCF (128, 96) = 2 x 2 x 2 x 2 x2 =
32
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By the method of division,
Hence, 32 is the HCF of 128 and 96.
9. Example 2: Two rods are 22 m and 26 m long. The rods are to be cut
into pieces of equal length. Find the maximum length of each piece.
• HCF or GCD is the required length of each piece.
• 22 = 2 x 11
• 26 = 2 x 13
• HCF or the greatest common divisor = 2
• Hence, the required maximum length of each piece is 2 m.
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10. Example 3: Find the greatest common factor of 24, 148 and
36.
• Prime factorisation of given numbers is
• 24 = 2 × 2 × 2 × 3
• 148 = 2 × 2 × 37
• 36 = 2 × 2 × 3 × 3
• Greatest common factor = 2 × 2 = 4
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