GSEB Class 6 Maths Solutions Chapter 3 Playing With Numbers Exercise 3.4

Question 1. Find the common factors of
(a) 20 and 28
(b) 15 and 25
(c) 35 and 50
(d) 56 and 120
Answer:
(a) 20 and 28
We have:
\( 20 = 1 \times 20 \)
\( 20 = 2 \times 10 \)
\( 20 = 4 \times 5 \)
All the factors of 20 are: 1, 2, 4, 5, 10 and 20.
Again,
\( 28 = 1 \times 28 \)
\( 28 = 2 \times 14 \)
\( 28 = 7 \times 4 \)
All the factors of 28 are: 1, 2, 4, 7, 14 and 28.
From these lists, the common factors of 20 and 28 are: 1, 2, and 4.
(b) 15 and 25
Since \( 15 = 1 \times 15 \)
\( 15 = 3 \times 5 \)
All the factors of 15 are: 1, 3, 5 and 15.
Again, \( 25 = 1 \times 25 \)
\( 25 = 5 \times 5 \)
All the factors of 25 are: 1, 5 and 25.
From these, the common factors of 15 and 25 are: 1 and 5.
(c) 35 and 50
Since \( 35 = 1 \times 35 \)
\( 35 = 5 \times 7 \)
All the factors of 35 are: 1, 5, 7 and 35.
Again, \( 50 = 1 \times 50 \)
\( 50 = 2 \times 25 \)
\( 50 = 5 \times 10 \)
All the factors of 50 are: 1, 2, 5, 10, 25, 50.
From these, the common factors of 35 and 50 are: 1 and 5.
(d) 56 and 120
Since \( 56 = 1 \times 56 \)
\( 56 = 2 \times 28 \)
\( 56 = 4 \times 14 \)
\( 56 = 7 \times 8 \)
All the factors of 56 are: 1, 2, 4, 7, 8, 14, 28 and 56.
Again,
\( 120 = 1 \times 120 \)
\( 120 = 2 \times 60 \)
\( 120 = 3 \times 40 \)
\( 120 = 4 \times 30 \)
\( 120 = 5 \times 24 \)
\( 120 = 6 \times 20 \)
\( 120 = 8 \times 15 \)
\( 120 = 10 \times 12 \)
All the factors of 120 are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and 120.
The common factors of 56 and 120 are: 1, 2, 4 and 8.
In simple words: To find common factors, list all the numbers that can divide each given number without a remainder. Then, check which numbers appear in every list. Those shared numbers are the common factors.

Exam Tip: When listing factors, start from 1 and go up, testing each number. Always pair factors (e.g., if 2 is a factor of 20, then 10 is also a factor). This helps ensure you don't miss any.

Question 2. Find the common factors of
(a) 4, 8 and 12
(b) 5, 15 and 25
Answer:
(a) 4, 8 and 12
Factors of 4 are: 1, 2 and 4.
Factors of 8 are: 1, 2, 4 and 8.
Factors of 12 are: 1, 2, 3, 4, 6 and 12.
The common factors of 4, 8 and 12 are: 1, 2 and 4.
(b) 5, 15 and 25
Factors of 5 are: 1 and 5.
Factors of 15 are: 1, 3, 5 and 15.
Factors of 25 are: 1, 5 and 25.
The common factors of 5, 15 and 25 are: 1 and 5.
In simple words: For three numbers, find all factors for each number. Then, find the numbers that show up in all three lists of factors. These are the shared factors.

Exam Tip: To avoid errors with multiple numbers, list the factors for each number separately and clearly. Then, carefully compare all the lists to identify the shared factors. It's often helpful to highlight or circle them.

Question 3. Find first three common multiples of
(a) 6 and 8
(b) 12 and 18
Answer:
(a) 6 and 8
Since multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, ...
Multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, ...
The first three common multiples of 6 and 8 are: 24, 48 and 72.
(b) 12 and 18
Since, multiples of 12 are: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, ...
Multiples of 18 are: 18, 36, 54, 72, 90, 108, 126, ...
The first three common multiples of 12 and 18 are: 36, 72 and 108.
In simple words: Write out the multiplication table for each number. Look for the numbers that appear in both lists. The first three numbers they share are the first three common multiples.

Exam Tip: Listing multiples up to a certain point is effective. For larger numbers or more common multiples, you can find the Least Common Multiple (LCM) first, and then multiply the LCM by 1, 2, 3, and so on to get subsequent common multiples.

Question 4. Write all the numbers less than 100 which are common multiples of 3 and 4.
Answer:
Multiples of 3, which are less than 100, are:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99.
Multiples of 4, which are less than 100, are:
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96.
Common multiples of 3 and 4, which are less than 100, are: 12, 24, 36, 48, 60, 72, 84 and 96.
In simple words: First, list all the numbers that 3 can divide into, up to 99. Then, list all the numbers that 4 can divide into, up to 99. Finally, find the numbers that appear in both of these lists; those are the common multiples.

Exam Tip: A number that is a common multiple of 3 and 4 must also be a multiple of their Least Common Multiple (LCM). The LCM of 3 and 4 is 12. So, you can simply list multiples of 12 to find the common multiples.

Question 5. Which of the following numbers are co-prime?
(a) 18 and 35
(b) 15 and 37
(c) 30 and 415
(d) 17 and 68
(e) 216 and 215
(f) 81 and 16
Answer:
(a) 18 and 35
Factors of 18 are: 1, 2, 3, 6, 9 and 18.
Factors of 35 are: 1, 5, 7 and 35.
Since their only common factor is 1, 18 and 35 are co-prime numbers.
(b) 15 and 37
Factors of 15 are: 1, 3, 5 and 15.
Factors of 37 are: 1 and 37.
Their only common factor is 1. Thus, 15 and 37 are co-prime numbers.
(c) 30 and 415
Factors of 30 are: 1, 2, 3, 5, 6, 10, 15 and 30.
Factors of 415 are: 1, 5, 83 and 415.
Their common factors are: 1 and 5.
Thus, 30 and 415 are not co-prime numbers because they share more than just 1 as a common factor.
(d) 17 and 68
Factors of 17 are: 1 and 17.
Factors of 68 are: 1, 2, 4, 17, 34 and 68.
Common factors of 17 and 68 are: 1 and 17.
Thus, 17 and 68 are not co-prime numbers.
(e) 216 and 215
Factors of 216 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108 and 216.
Factors of 215 are: 1, 5, 43 and 215.
The common factor of 216 and 215 is 1.
Thus, 216 and 215 are co-prime numbers.
(f) 81 and 16
Factors of 81 are: 1, 3, 9, 27 and 81.
Factors of 16 are: 1, 2, 4, 8 and 16.
The common factor of 81 and 16 is 1.
Thus, 81 and 16 are co-prime numbers.
In simple words: Two numbers are co-prime if their only shared factor is the number 1. You need to list all the numbers that can divide each number. If 1 is the only number found in both lists, then they are co-prime.

Exam Tip: To quickly check if numbers are co-prime, find the Greatest Common Divisor (GCD) or Highest Common Factor (HCF). If the GCD is 1, the numbers are co-prime. If the GCD is greater than 1, they are not co-prime.

Question 6. A number is divisible by both 5 and 12. By which other number will that number be always divisible?
Answer: The given number will always be divisible by the product of 5 and 12. Therefore, the number will be divisible by \( 5 \times 12 \), which is 60.
In simple words: If a number can be divided by two different numbers (like 5 and 12), then it can also be divided by what you get when you multiply those two numbers together. So, it will always be divisible by 60.

Exam Tip: Remember that if a number is divisible by two co-prime numbers, it is also divisible by their product. In this case, 5 and 12 are co-prime, so the rule applies directly.

Question 7. A number is divisible by 12. By what other numbers will that number be divisible?
Answer: If a number is divisible by 12, it will also be divisible by all the factors of 12. The factors of 12 are 1, 2, 3, 4, 6 and 12. Therefore, the number will be divisible by 1, 2, 3, 4 and 6, in addition to 12 itself.
In simple words: If a number can be divided by 12, it also means that number can be divided by all the smaller numbers that can divide 12. These smaller numbers are called factors of 12.

Exam Tip: When a number is divisible by another number, it is also divisible by every factor of that divisor. This is a fundamental property of divisibility that is often tested in number theory questions.