GSEB Class 6 Maths Solutions Chapter 3 Playing With Numbers Exercise 3.6

Question 1. Find the HCF of the following numbers:
(a) 18, 48
(b) 30, 42
(c) 18, 60
(d) 27, 63
(e) 36, 84
(f) 34, 102
(g) 70, 105, 175
(h) 91, 112, 49
(i) 18, 54, 81
(j) 12, 45, 75
Answer:
(a) The numbers that divide 18 are: 1, 2, 3, 6, 9, and 18.
The numbers that divide 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
The factors they both share are: 1, 2, 3, and 6.
As the biggest shared factor is 6, the HCF for 18 and 48 equals 6.

Exam Tip: Remember to list all factors carefully and identify the largest common one to find the HCF correctly.

Answer:
(b) The numbers that divide 30 are: 1, 2, 3, 5, 6, 10, 15, and 30.
The numbers that divide 42 are: 1, 2, 3, 6, 7, 14, 21, and 42.
The factors they both share are: 1, 2, 3, and 6.
As the biggest shared factor is 6, the HCF for 30 and 42 equals 6.

Exam Tip: To avoid errors, always list the factors in increasing order for each number.

Answer:
(c) The numbers that divide 18 are: 1, 2, 3, 6, 9, and 18.
The numbers that divide 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.
The factors they both share are: 1, 2, 3, and 6.
As the biggest shared factor is 6, the HCF for 18 and 60 equals 6.

Exam Tip: Double-check your lists of factors to ensure no common factors are missed.

Answer:
(d) The numbers that divide 27 are: 1, 3, 9, and 27.
The numbers that divide 63 are: 1, 3, 7, 9, 21, and 63.
The factors they both share are: 1, 3, and 9.
As the biggest shared factor is 9, the HCF for 27 and 63 equals 9.

Exam Tip: Identifying prime factors first can simplify finding all factors of a number.

Answer:
(e) The numbers that divide 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.
The numbers that divide 84 are: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.
The factors they both share are: 1, 2, 3, 4, 6, and 12.
As the biggest shared factor is 12, the HCF for 36 and 84 equals 12.

Exam Tip: When dealing with larger numbers, prime factorization can be a more efficient method for finding the HCF.

Answer:
(f) The numbers that divide 34 are: 1, 2, 17, and 34.
The numbers that divide 102 are: 1, 2, 3, 6, 17, 34, 51, and 102.
The factors they both share are: 1, 2, 17, and 34.
As the biggest shared factor is 34, the HCF for 34 and 102 equals 34.

Exam Tip: Numbers that are multiples of each other will have the smaller number as their HCF.

Answer:
(g) The numbers that divide 70 are: 1, 2, 5, 7, 10, 14, 35, and 70.
The numbers that divide 105 are: 1, 3, 5, 7, 15, 21, 35, and 105.
The numbers that divide 175 are: 1, 5, 7, 25, 35, and 175.
The factors they all share are: 1, 5, 7, and 35.
As the biggest shared factor is 35, the HCF for 70, 105, and 175 equals 35.

Exam Tip: For three or more numbers, find factors common to all of them, not just pairs.

Answer:
(h) The numbers that divide 91 are: 1, 7, 13, and 91.
The numbers that divide 112 are: 1, 2, 4, 7, 8, 14, 16, 28, 56, and 112.
The numbers that divide 49 are: 1, 7, and 49.
The factors they all share are: 1 and 7.
As the biggest shared factor is 7, the HCF for 91, 112, and 49 equals 7.

Exam Tip: Always make sure to consider all given numbers when finding their common factors.

Answer:
(i) The numbers that divide 18 are: 1, 2, 3, 6, 9, and 18.
The numbers that divide 54 are: 1, 2, 3, 6, 9, 18, 27, and 54.
The numbers that divide 81 are: 1, 3, 9, 27, and 81.
The factors they all share are: 1, 3, and 9.
As the biggest shared factor is 9, the HCF for 18, 54, and 81 equals 9.

Exam Tip: Practicing finding factors of various numbers helps improve speed and accuracy in HCF calculations.

Answer:
(j) The numbers that divide 12 are: 1, 2, 3, 4, 6, and 12.
The numbers that divide 45 are: 1, 3, 5, 9, 15, and 45.
The numbers that divide 75 are: 1, 3, 5, 15, 25, and 75.
The factors they all share are: 1 and 3.
As the biggest shared factor is 3, the HCF for 12, 45, and 75 equals 3.
In simple words: To get the HCF, first list every number that divides each of the numbers you're given. Then, find the largest number that shows up in all of those lists. That big shared number is your HCF.

Exam Tip: Listing factors correctly is the foundation; a single missed factor can lead to an incorrect HCF.

Question 2. What is the HCF of two consecutive
(a) numbers ?
(b) even numbers ?
(c) odd numbers ?
Answer:
(a) The common factor of any two numbers that follow each other is always 1.
The HCF of two consecutive numbers is 1.
(b) The common factors of two consecutive even numbers are 1 and 2.
The HCF of two consecutive even numbers is 2.
(c) The common factor of two consecutive odd numbers is 1.
The HCF of two consecutive odd numbers is 1.
In simple words: For numbers right next to each other, the biggest shared divider is usually small. For any two consecutive numbers, it's 1. For two even numbers in a row, it's 2. For two odd numbers in a row, it's 1 again.

Exam Tip: Remember these special cases for HCFs of consecutive numbers as they are common short-answer questions.

Question 3. HCF of co-prime numbers 4 and 15 was found as follows by factorisation : \( 4 = 2 \times 2 \) and \( 15 = 3 \times 5 \) since there is no common prime factor, so HCF of 4 and 15 is 0. Is the answer correct ? If not, what is the correct HCF?
Answer: No number can be a factor of zero.
The number 1 is a factor of every number.
If there are no common prime factors, it means that 1 is the common factor.
The given answer is not correct. The correct answer is: HCF of 4 and 15 = 1.
In simple words: The given answer is wrong because the HCF can never be zero. Even if numbers don't share any prime factors, they always share 1 as a common factor. So, the HCF for 4 and 15 is actually 1.

Exam Tip: Always remember that the HCF of any set of numbers, especially co-prime numbers, can never be 0. It is always at least 1.