Class 9 Maths Ganita Manjari Chapter 11 Exercise 11.2 Solutions The World of Algorithms
The Class 9 Maths Ganita Manjari Chapter 11 Exercise 11.2 Solutions – The World of Algorithms help students understand how algorithms can be used to compare lists, find least common multiples, and identify divisors efficiently. The exercise focuses on developing logical thinking through step-by-step procedures and shows how mathematical tasks can be solved systematically using clear rules.
The solutions are explained in a simple and structured manner to help students understand each algorithm and apply it correctly. Students can also access the Class 9 Maths Ganita Manjari Chapter 11 Exercise 11.2 Solutions PDF for easy revision and offline study.
NCERT Class 9 Maths Ganita Manjari Chapter 11 Exercise Set 11.2
1. Suppose List 1 and List 2 are two lists of numbers in increasing order.
(i) Write an algorithm to find elements in List 1 that are not present in List 2.
(ii) Write an algorithm to find elements in List 2 that are not present in List 1.
Answer:
(i) Write an algorithm to find elements in List 1 that are not present in List 2.
Solution:
- Start with an empty list called only-in-1.
- Start from the first element of List 1 and the first element of List 2.
- Compare the two elements.
- If both elements are equal, move to the next element in both lists.
- If the element in List 1 is smaller, add it to only-in-1 and move to the next element in List 1.
- If the element in List 2 is smaller, move to the next element in List 2.
- Repeat these steps until one of the lists ends.
- If List 2 ends first, add all the remaining elements of List 1 to only-in-1.
- Report only-in-1.
(ii) Write an algorithm to find elements in List 2 that are not present in List 1.
Solution:
- Start with an empty list called only-in-2.
- Start from the first element of List 1 and the first element of List 2.
- Compare the two elements.
- If both elements are equal, move to the next element in both lists.
- If the element in List 1 is smaller, move to the next element in List 1.
- If the element in List 2 is smaller, add it to only-in-2 and move to the next element in List 2.
- Repeat these steps until one of the lists ends.
- If List 1 ends first, add all the remaining elements of List 2 to only-in-2.
- Report only-in-2.
2. Describe an algorithm to compute the least common multiple (lcm) of two numbers.
Answer:
To find the least common multiple (LCM) of two positive numbers:
- Let the two numbers be a and b.
- Start with the larger of the two numbers.
- Check whether this number is divisible by both a and b.
- If it is divisible by both, it is the LCM.
- If not, move to the next multiple of the larger number.
- Keep checking successive multiples until one is divisible by both numbers.
- Report that number as the LCM.
Example: Find the LCM of 4 and 6.
Multiples of 6 are: 6, 12, 18, 24, …
12 is the first multiple of 6 that is also divisible by 4.
Therefore, LCM(4, 6) = 12.
3. Divisors occur in pairs. For instance, the divisors of 18 are (1, 18), (2, 9) and (3, 6).
(i) If we write out divisors in pairs, how many numbers do we have to examine between 1 and n to find all the divisors of n?
(ii) If we list out the divisors in pairs, will our gcd algorithm still work in the manner we have described?
(i) If we write out divisors in pairs, how many numbers do we have to examine between 1 and n to find all the divisors of n?
Answer:
Divisors of n occur in pairs. If d is a divisor of n, then n ÷ d is also a divisor of n, and their product is n.
Therefore, we do not need to check every number from 1 to n. We only need to examine the integers from 1 up to √n. After that, the divisors are simply the partners of the divisors already found.
For example, for n = 18:
√18 ≈ 4.24
So, we need to check only 1, 2, 3 and 4.
We find that 1, 2 and 3 divide 18, giving the pairs:
(1, 18), (2, 9), (3, 6)
Therefore, to find all the divisors of n, we need to examine only the integers up to √n, that is, ⌊√n⌋ numbers.
(ii) If we list out the divisors in pairs, will our gcd algorithm still work in the manner we have described?
Answer:
The divisors themselves remain the same when they are written in pairs, so the greatest common divisor does not change.
However, if the gcd algorithm we described requires the divisors to be listed in increasing order, then the divisor pairs must first be arranged appropriately before applying that algorithm.
For example, the divisors of 18 written as pairs are:
(1, 18), (2, 9), (3, 6)
These contain exactly the same divisors:
1, 2, 3, 6, 9, 18
Thus, listing divisors in pairs does not affect which divisors are common to the two numbers or what their GCD is. The algorithm will still work as long as the divisors are compared in the order required by the algorithm.
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