RBSE Solutions Class 6 Maths Chapter 2 Relation Among Numbers Exercise 2.3

Question 1. Find out H.C.F. of the following numbers:
(i) 36, 84
(ii) 28, 42
(iii) 13, 26, 52
(iv) 15, 35, 40
(v) 23, 31, 93
Answer:
(i) Factors of \( 36 = 2 \times 2 \times 3 \times 3 \)
Factors of \( 84 = 2 \times 2 \times 3 \times 7 \)
H.C.F. of 36 and 84 \( = 2 \times 2 \times 3 = 12 \)
(ii) Factors of \( 28 = 2 \times 2 \times 7 \)
Factors of \( 42 = 2 \times 3 \times 7 \)
H.C.F. of 28 and 42 \( = 2 \times 7 = 14 \)
(iii) Factors of \( 13 = 1 \times 13 \)
Factors of \( 26 = 2 \times 13 \)
Factors of \( 52 = 2 \times 2 \times 13 \)
H.C.F. of 13, 26 and 52 \( = 13 \)
(iv) Factors of \( 15 = 3 \times 5 \)
Factors of \( 35 = 5 \times 7 \)
Factors of \( 40 = 2 \times 2 \times 2 \times 5 \)
H.C.F. of 15, 35 and 40 \( = 5 \)
(v) Factors of \( 23 = 1 \times 23 \)
Factors of \( 31 = 1 \times 31 \)
Factors of \( 93 = 1 \times 3 \times 31 \)
H.C.F. of 23, 31 and 93 \( = 1 \)
In simple words: To find the H.C.F., list all the prime factors for each number. Then, multiply only the common prime factors that appear in every list. This is the largest number that divides all of them without a remainder.

🎯 Exam Tip: Remember that 1 is the H.C.F. for any set of numbers that do not share any common prime factors (like prime numbers).

Question 2. Find out H.C.F. of the following:
(i) Two successive numbers
(ii) Two successive even numbers
(iii) Two successive odd numbers
Answer:
(i) Two successive numbers, like 1 and 2, or 5 and 6, always have an H.C.F. of 1. This is because they do not share any common prime factors.
(ii) Two successive even numbers, like 2 and 4, or 10 and 12, always have an H.C.F. of 2. They always share 2 as a common factor.
(iii) Two successive odd numbers, like 1 and 3, or 7 and 9, always have an H.C.F. of 1. For example, factors of \( 1 = 1 \times 1 \), factors of \( 3 = 1 \times 3 \). Their only common factor is 1.
In simple words: The H.C.F. for numbers that come right after each other (successive) is always 1. For even numbers that come right after each other, the H.C.F. is 2. For odd numbers that come right after each other, the H.C.F. is 1.

🎯 Exam Tip: When dealing with H.C.F. questions about types of numbers (successive, even, odd), always test with a couple of small examples to confirm the pattern.

Question 3. Width and length of the floor is 25 meters and 30 meters respectively. Find out the length of the longest rope which can be use to measure length and width of the room exactly.
Answer:
The breadth of the floor is 25 m.
The length of the floor is 30 m.
To find the longest rope that can measure both exactly, we need to find the H.C.F. of 25 and 30.
Factors of \( 25 = 5 \times 5 \)
Factors of \( 30 = 2 \times 3 \times 5 \)
The common factor is 5.
So, the H.C.F. of 25 and 30 is 5.
Thus, the length of the longest rope will be 5 m.
In simple words: We need to find the largest number that divides both 25 and 30 without a remainder. This number is 5, so a 5-meter rope is the longest one that works perfectly.

🎯 Exam Tip: Questions asking for the "longest," "greatest," or "maximum" measurement that can divide given quantities typically require finding the Highest Common Factor (H.C.F.).

Question 4. Three oil tankers are of capacity 96 litres, 100 litres and 144 liters. Find out maximum measurements to measure the oil of all three tankers exactly.
Answer:
The capacity of the first oil tanker is 96 litres.
The capacity of the second oil tanker is 100 litres.
The capacity of the third oil tanker is 144 litres.
To find the maximum measurement that can measure all three tankers exactly, we need to find the H.C.F. of 96, 100, and 144.
Factors of \( 96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \)
Factors of \( 100 = 2 \times 2 \times 5 \times 5 \)
Factors of \( 144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \)
The common prime factors for all three numbers are \( 2 \times 2 \).
So, the H.C.F. of 96, 100, and 144 \( = 2 \times 2 = 4 \).
Thus, the required maximum measure will be 4 litres. This means a 4-liter container can perfectly measure the oil from all three tankers without any leftovers.
In simple words: We want to find the biggest size of a measuring can that can empty all three oil tankers perfectly. By finding the H.C.F. of their capacities, we found that a 4-liter can is the correct size.

🎯 Exam Tip: Always list out the prime factors carefully for each number and identify only the factors that are common to all numbers to find the H.C.F. correctly.

Question 5. Find out the length of the longest rope to measure distances of 36 meters, 54 meters, and 90 meters.
Answer:
The given distances are 36 meters, 54 meters, and 90 meters.
To find the length of the longest rope that can measure all these distances exactly, we need to find the H.C.F. of 36, 54, and 90.
Factors of \( 36 = 2 \times 2 \times 3 \times 3 \)
Factors of \( 54 = 2 \times 3 \times 3 \times 3 \)
Factors of \( 90 = 2 \times 3 \times 3 \times 5 \)
The common prime factors for all three numbers are \( 2 \times 3 \times 3 \).
So, the H.C.F. of 36, 54, and 90 \( = 2 \times 3 \times 3 = 18 \).
Thus, the length of the longest rope will be 18 m. This 18-meter rope can be used to measure each distance without any leftover length.
In simple words: We need to find the biggest length of rope that can measure 36m, 54m, and 90m perfectly. The H.C.F. of these numbers is 18, so an 18-meter rope is the longest one that works.

🎯 Exam Tip: When the problem asks for a common measuring unit or dimension that fits multiple given values exactly, H.C.F. is the key mathematical operation to use.