The GCD (Greatest Common Divisor), also known as the HCF (Highest Common Factor), of two or more numbers is the largest number that divides all of them exactly, without leaving a remainder.
Key Points:
- It is the greatest number that is a factor of each of the numbers.
- For two numbers a and b, the GCD is the largest number d such that both a ÷ d and b ÷ d are integers.
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Solved Questions on Greatest Common Divisor (Medium)
Example 1: A school has 42 boys and 70 girls who need to be grouped to make teams. Find the maximum number of groups so that boys and girls are equally distributed among each group also find the number of boys and girls in a group.
Answer:
To find the maximum possible number of group, we need to find the HCF of 42 and 70.
Prime factors of 42:
2 \times 3 \times 7
Prime factors of 70:2 \times 5 \times 7 Common factors: 2 \times 7 = 14
The HCF is 14, so a maximum of 14 groups can be formed.
Number of students in each groups:
Boys: 42 ÷ 14 = 3.
Girls: 70 ÷ 14 = 5.
Maximum 14 groups can be formed with each group 3 boys and 5 girls.
Example 2: If the HCF of two numbers is 1, what can you conclude about those numbers?
Answer:
When the HCF of two numbers is 1, it indicates that these two numbers are coprime . As Coprime numbers are integers that share no common factors other than 1. In other words, their HCF is the smallest possible, which is 1. This implies that the numbers have no common factors except for unity, making them mutually prime to each other. For example, 5 and 8 are coprime because their only common factor is 1.
Example 3: Can the HCF of two prime numbers be a prime number other than 1? Explain.
Answer:
The HCF (Highest Common Factor) of two prime numbers cannot be a prime number other than 1.
Consider the scenario where the HCF of two numbers is a prime number other than 1. Let's say the HCF is 'p', where 'p' is a prime number greater than 1.
- If 'p' is the HCF, it means 'p' is a common factor of the two numbers.
- However, since 'p' is a prime number greater than 1, it cannot be divided by any other number except 1 and itself.
- This implies that the two numbers can only be divided by 'p' and nothing else.
- But this contradicts the definition of HCF because the HCF is supposed to be the largest number that can exactly divide both of them. If 'p' is the HCF, it should be the largest, but it cannot be because it cannot be divided by any number other than 1 and itself.
Therefore, the HCF of two numbers cannot be a prime number other than 1. It must always be 1 or a composite number (a number with more than two factors).
Example 4: Sarah has 12 apples and 16 oranges, and she wants to arrange them into equal-sized groups. She wants to ensure that no fruits are left over in each group. What is the largest number of apples and oranges she can put in each group?
Answer:
To find the largest number of groups of apples and oranges that Sarah can make without any fruits left over, we need to calculate the HCF of the number of apples and number of oranges she has, which is 12 and 16.
12 = 2 x 2 x 3 x 3
16 = 2 x 2 x 2 x 2HCF ( 12, 16) = 2 x 2 = 4
The HCF of 12 and 16 is 4, so Sarah can make 4 groups with 3 apples and 4 oranges in each group, and no fruits will be left over.
Example 5. Sam is preparing dinner plates. She has 60 pieces of momos and 8 rolls. If she wants to make all the plates identical without any food left over, what is the greatest number of plates Sarah can prepare ?
Answer:
In order for all the plates to look identical with the highest no of plates, we need to find the HCF of 8 rolls and 60 pieces of momos
HCF (8,60) = 4
Hence, Sam can prepare a total of 4 plates with 15 momos and 2 rolls in each plate.
Example 6. A juice seller has three different types of fruit juices: apple juice, orange juice, and grape juice. He has 403 liters of apple juice, 434 liters of orange juice, and 465 liters of grape juice. What is the minimum number of identical containers he needs to store each type of juice separately without mixing them?
Answer:
For the minimum number of containers of equal size, the size of each container must be of the greatest volume.
To get the greatest volume of each container, we need to find HCF of 403, 434 and 465.H.C.F (403, 434, 465) = 31 liters
Each container must be of the volume 31 liters.Number of containers required are = (403/31) + (434/31) + (465/31) = 42
Hence, the minimum number of containers required are 42.
Example 7. Emma has 48 math books and 64 science books. She wants to arrange them into stacks so that each stack has the same number of books, and each stack only contains one type of book. What is the maximum number of books in each stack?
Answer:
To arrange books in the largest possible equal stacks, we need to find the HCF of 48 and 64.
Prime factors of 48: 23 x 3
Prime factors of 64: 26
Common factors: 24 = 16The HCF is 16, so each stack can have a maximum of 16 books.
Math books: 48 ÷ 16 = 3 stacks.
Science books: 64 ÷ 16 = 4 stacks.The maximum number of books in each stack is 16, resulting in 7 stacks in total.
Also Read : Short Tricks to solve HCF
GCD Practice Problems (Medium)
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GCD Practice Questions (Easy Level)
GCD Practice Questions (hard Level)