GSEB Class 6 Maths Solutions Chapter 3 Playing With Numbers InText Questions

Try These (Page 48)

Question 1. Find the possible factors of 45, 30 and 36.
(i) Factors of 45:
Answer: We get: \( 45 = 1 \times 45 \)
\( 45 = 3 \times 15 \)
\( 45 = 5 \times 9 \)
So, the factors of 45 include: 1, 3, 5, 9, 15, and 45.
(ii) Factors of 30:
\( 30 = 1 \times 30 \)
\( 30 = 2 \times 15 \)
\( 30 = 3 \times 10 \)
\( 30 = 5 \times 6 \)
The factors for 30 are: 1, 2, 3, 5, 6, 10, 15, and 30.
(iii) Factors of 36:
\( 36 = 1 \times 36 \)
\( 36 = 2 \times 18 \)
\( 36 = 3 \times 12 \)
\( 36 = 4 \times 9 \)
\( 36 = 6 \times 6 \)
Thus, the factors of 36 include: 1, 2, 3, 4, 6, 9, 12, 18, and 36.
In simple words: To find factors, list all pairs of numbers that multiply to give the original number. Include 1 and the number itself. Combine all these unique numbers to get the complete list of factors.

Exam Tip: Remember to list factors in ascending order and ensure you have included both 1 and the number itself in the list.

Try These (Page 52)

Question 1. Observe that \( 2 \times 3 + 1 = 7 \) is a prime number. Here, 1 has been added to a multiple of 2 to get a prime number. Can you find some more numbers of this type?
Answer: We can find several examples:
\( 2 \times 2 + 1 = 5 \), which is a prime number.
\( 2 \times 5 + 1 = 11 \), which is a prime number.
\( 2 \times 6 + 1 = 13 \), which is a prime number.
\( 2 \times 8 + 1 = 17 \), which is a prime number.
\( 2 \times 9 + 1 = 19 \), which is a prime number.
\( 2 \times 10 + 1 = 23 \), which is a prime number.
In simple words: You can make new prime numbers by taking a multiple of 2 and adding 1. Try different numbers in place of N to see which ones work out as prime.

Exam Tip: When looking for prime numbers, always check if the number has any divisors other than 1 and itself. If it does, it's not prime.

Try These (Page 58)

Question 1. Find the common factors of
(a) 8, 20
(b) 9, 15
Answer:
(a) 8, 20
For part (a), we have:
\( 8 = 1 \times 8 \)
\( 8 = 2 \times 4 \)
So, all the factors of 8 are: 1, 2, 4, and 8. (i)
Next, for 20:
\( 20 = 1 \times 20 \)
\( 20 = 2 \times 10 \)
\( 20 = 4 \times 5 \)
The factors of 20 are: 1, 2, 4, 5, 10, and 20. (ii)
Based on (i) and (ii), the common factors of 8 and 20 are: 1, 2, and 4.
(b) 9, 15
For part (b), we observe:
\( 9 = 1 \times 9 \)
\( 9 = 3 \times 3 \)
Thus, the factors of 9 are: 1, 3, and 9. (i)
Similarly, for 15, we have: \( 15 = 1 \times 15 \)
\( 15 = 3 \times 5 \)
The factors of 15 are: 1, 3, 5, and 15.
Considering (i) and (ii), the common factors of 9 and 15 are: 1 and 3.
In simple words: First, find all the factors for each number. Then, look for the numbers that appear in all of the lists. Those are the common factors.

Exam Tip: To ensure you don't miss any common factors, list all factors of each number systematically before comparing them.

Try These (Page 61)

Question 1. Write the prime factorisations of 16, 28, 38.
Answer:
(i) 16
For 16, we perform prime factorization:
2|16
2|8
2|4
2|2
|1
The prime factorization of \( 16 \) is \( 2 \times 2 \times 2 \times 2 \).
(ii) 28
For 28, the prime factors are found as follows:
2|28
2|14
7|7
|1
So, the prime factorization of \( 28 \) is \( 2 \times 2 \times 7 \).
(iii) 38
For 38, we get:
2|38
19|19
|1
The prime factorization of \( 38 \) is \( 2 \times 19 \).
In simple words: Break down a number into its smallest prime building blocks. Keep dividing by prime numbers until you reach 1. The primes you used are the prime factors.

Exam Tip: Always start dividing by the smallest prime number (2) and continue until the number is no longer divisible, then move to the next prime (3, 5, 7, etc.).

Try These (Page 63)

Question 1. Find the HCF of the following:
(i) 24 and 36
(ii) 15, 25 and 30
(iii) 8 and 12
(iv) 12, 16 and 28
Answer:
(i) 24 and 36
The factors for 24 are: 1, 2, 3, 4, 6, 12, and 24.
The factors for 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.
Common factors include: 1, 2, 3, 4, 6, and 12.
Given these common factors, the highest among them is 12.
Thus, the HCF of 24 and 36 is 12.
(ii) 15, 25 and 30
The factors of 15 are: 1, 3, 5, and 15.
The factors of 25 are: 1, 5, and 25.
The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, and 30.
The common factors identified are: 1 and 5.
As a result, the highest common factor is 5.
Therefore, the HCF of 15, 25, and 30 is 5.
(iii) 8 and 12
The factors for 8 are: 1, 2, 4, and 8.
The factors for 12 are: 1, 2, 3, 4, 6, and 12.
The common factors include: 1, 2, and 4.
Given these, the highest common factor is 4.
Thus, the HCF of 8 and 12 is 4.
(iv) 12, 16 and 28
The factors of 12 are: 1, 2, 3, 4, 6, and 12.
The factors of 16 are: 1, 2, 4, 8, and 16.
The factors of 28 are: 1, 2, 4, 7, 14, and 28.
The common factors found are: 1, 2, and 4.
Therefore, the highest common factor is 4.
So, the HCF of 12, 16, and 28 is 4.
In simple words: Find all the factors for each number. Then, pick out all the factors that they share. The biggest number among these shared factors is the Highest Common Factor.

Exam Tip: When finding HCF for multiple numbers, ensure you list all factors for each number accurately, then identify only those factors present in every list.