Maharashtra Board Class 5 Maths Chapter 8 Multiples and Factors Set 35 Solutions

Question 1. 22, 24
Answer: Common factors of 22 and 24 are 1 and 2. (Not only 1 common factor) So, 22, 24 are not co-prime numbers.
In simple words: Co-prime numbers share only 1 as a common factor. Since 22 and 24 share 1 and 2, they are not co-prime.

🎯 Exam Tip: To determine if numbers are co-prime, always find all common factors. If the only common factor is 1, they are co-prime.

Question 2. 14, 21
Answer: Common factors of 14 and 21 are 1 and 7. So, this pair is not co-prime numbers.
In simple words: For numbers to be co-prime, their only common factor must be 1. Since 14 and 21 also share 7 as a factor, they are not co-prime.

🎯 Exam Tip: Clearly list all factors for each number to correctly identify common factors. Missing one factor can lead to an incorrect conclusion.

Question 3. 10, 33
Answer: Common factors of 10 and 33 is only 1. So, 10 and 33 are co-prime numbers.
In simple words: The only number that divides both 10 and 33 without a remainder is 1, making them co-prime.

🎯 Exam Tip: Remember that co-prime numbers don't have to be prime themselves; they just need to share only 1 as a common factor.

Question 4. 11, 30
Answer: Common factors of 11 and 30 is only 1. So, 11 and 30 are co-prime numbers.
In simple words: As 11 is a prime number and 30 is not a multiple of 11, their only common factor is 1, so they are co-prime.

🎯 Exam Tip: When one of the numbers is prime, check if the other number is a multiple of that prime number. If not, they are likely co-prime.

Question 5. 5, 7
Answer: Common factor of 5 and 7 is only 1. So, 5 and 7 are co-prime numbers.
In simple words: Both 5 and 7 are prime numbers, and prime numbers always have only 1 as a common factor with another prime (unless they are the same prime). Thus, they are co-prime.

🎯 Exam Tip: Any two distinct prime numbers are always co-prime to each other.

Question 6. 15, 16
Answer: Common factors of 15 and 16 is only 1. So, 15 and 16 are co-prime numbers.
In simple words: Despite being consecutive numbers, 15 and 16 share only the factor 1, making them co-prime.

🎯 Exam Tip: Consecutive integers are always co-prime because their only common factor is 1.

Question 7. 50, 52
Answer: Common factors of 50 and 52 are 1 and 2. So, 50 and 52 are not co-prime numbers.
In simple words: Both 50 and 52 are even numbers, meaning they are both divisible by 2, so they cannot be co-prime.

🎯 Exam Tip: If two numbers are both even, they will always have 2 as a common factor, meaning they can never be co-prime.

Question 8. 17, 18
Answer: Common factors of 17 and 18 is only 1. So, 17 and 18 are co-prime numbers.
In simple words: 17 is a prime number, and 18 is not a multiple of 17, so their only common factor is 1, making them co-prime.

🎯 Exam Tip: Use primality tests or simple division to quickly find factors. For prime numbers, checking factors is simpler.

Activity 1:

• Write numbers from 1 to 60.
• Draw a blue circle around multiples of 2.
• Draw a red circle around multiples of 4.
• Do all numbers with a blue circle also have a red circle around them?
• Do all the numbers with a red circle have a blue circle around them?
• Are all multiples of 2 also multiples of 4?
• Are all multiples of 4 also multiples of 2?

Activity 2 :

• Write numbers from 1 to 60.

• Draw a triangle around multiples of 2.
• Draw a circle around multiples of 3.
• Now find numbers divisible by 6. Can you find a property that they share?

Eratosthenes' Method Of Finding Prime Numbers

Eratosthenes was a mathematician who lived in Greece about 250 BC. He discovered a method to find prime numbers. It is called Eratosthenes' Sieve. Let us see how to find prime numbers between 1 and 100 with this method.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र 1 से 100 तक की संख्याओं को एक ग्रिड में दर्शाता है। यह एराटोस्थनीज की छलनी विधि का प्रतिनिधित्व करता है, जिसमें अभाज्य संख्याओं को वृत्त करके और उनके गुणजों को काटकर अभाज्य संख्याएँ (प्राइम नंबर्स) ज्ञात की जाती हैं। छात्र इस ग्रिड का उपयोग करके 1 से 100 के बीच की अभाज्य संख्याओं की पहचान कर सकते हैं।

• 1 is neither a prime nor a composite number. Put a square [ ] around it
• 2 is a prime number, so put a circle around it.
• Next, strike out all the multiples of 2. This tells us that of these 100 numbers more than half of numbers are not prime numbers.
• The first number after 2 not yet struck off is 3. So, 3 is a prime number.
• Draw a circle around 3. Strike out all the multiples of 3.
• The next number after 3 not struck off yet is 5. So, 5 is a prime number.
• Draw a circle around 5. Put a line through all the multiples of 5.
• The next number after 5 without a line through it is 7. So, 7 is a prime number.
• Draw a circle around 7. Put a line through all the multiples of 7.

In this way, every number between 1 and 100 will have either a circle or a line through it. The circled numbers are prime numbers. The numbers with a line through them are composite numbers.

One More Method To Find Prime Numbers

See how numbers from 1 to 36 have been arranged in six columns in the table alongside. Continue in the same way and write numbers up to 102 in these six columns. You will see that, in the columns for 2, 3, 4, and 6, all the numbers are composite numbers except for the prime numbers 2 and 3. This means that all the remaining prime numbers will be in the columns for 1 and 5. Now isn't it easier to find them? So, go ahead, find the prime numbers!

Something More

• Prime numbers with a difference of two are called twin prime numbers. Some twin prime number pairs are 3 and 5, 5 and 7, 29 and 31 and 71 and 73. 5347421 and 5347423 are also a pair of twin prime numbers.
• There are eight pairs of twin prime numbers between 1 and 100. Find them.
• Euclid the mathematician lived in Greece about 300 BC. He proved that if prime numbers, 2, 3, 5, 7, ......., are written in serial order, the list will never

end, meaning that the number of prime numbers is infinite.

Multiples And Factors Problem Set 35 Additional Important Questions And Answers

Determine whether the pairs of numbers given below are co-prime numbers.

Question 1. (12,18)
Answer: Common factors of 12 and 18 are 1, 2, 3, 6. Hence 12 and 18 are not co-prime numbers.
In simple words: Since 12 and 18 share common factors other than 1 (like 2, 3, and 6), they are not co-prime.

🎯 Exam Tip: Always list all factors for each number to ensure you don't miss any common ones, especially for larger numbers.

Question 2. (26, 39)
Answer: Common factors of 26 and 39 are 1 and 13. Hence, 26 and 39 are not co-prime numbers.
In simple words: Both 26 and 39 are divisible by 13, meaning 13 is a common factor besides 1, so they are not co-prime.

🎯 Exam Tip: For numbers that don't seem obviously divisible, try checking for prime factors to find common factors more easily.

Question 3. (23, 29)
Answer: Common factor of 23 and 29 is only 1. Hence, 23 and 29 are co-prime numbers.
In simple words: Both 23 and 29 are prime numbers, and since they are different, their only common factor is 1, making them co-prime.

🎯 Exam Tip: If two numbers are distinct prime numbers, they are always co-prime.

Question 4. (28, 32)
Answer: Common factors of 28 and 32 are 1, 2, 4 (not only 1). Hence, 28, 32 are not co-prime numbers.
In simple words: Since 28 and 32 are both even, they share 2 as a common factor. They also share 4, therefore, they are not co-prime.

🎯 Exam Tip: If numbers have multiple common factors (e.g., 2 and 4), ensure you list all of them to demonstrate full understanding.