Class 9 Mathematics

Chapter 11 Solutions — The World of Algorithms

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Solutions

Overview

Step-by-step NCERT solutions for The World of Algorithms (Chapter 11, NCERT Class 9 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the The World of Algorithms textbook chapter.

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What these solutions cover

All 15 questions in The World of Algorithms are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise Set 11.1

Add two 4-digit numbers using the steps we have written down. Make sure you follow the steps precisely; do not perform any action that is not explicitly mentioned. Are you able to obtain the correct result?
What happens if you add a 5-digit number to a 3-digit number. Do our steps handle this situation correctly?
Why is it important to align the columns from right to left?
In Step 3, why cannot the value of carry be more than 1?
What happens if we do not include the fifth step in the algorithm above? Give examples where the algorithm will work correctly and where it will fail to work.

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.
Describe an algorithm to compute the least common multiple (lcm) of two numbers.
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?

Exercise Set 11.3

How would our original algorithm change if we computed the divisors of n by examining the numbers from 1 to n in reverse order, from n down to 1?
What about the last algorithm described above? What happens when we look at common divisors starting from min(m, n) and work backwards to 1?

End-of-Chapter Exercises

1.
Compute the following using the improved version of Euclid's algorithm (Āryabhaṭa's division algorithm).
- (i) gcd(375, 825)
- (ii) gcd(51000, 81000)
- (iii) gcd(1789287, 237656)
- (iv) gcd(2587392, 157656)
Assume that m ≥ n. Verify that d divides m and n if and only if d divides both n and m mod n.
Write an algorithm prime(n) to check if n is prime. (Hint: A prime number p has exactly two distinct factors, 1 and p. Can you make use of divisors(n) to write out prime(n)?)
(*) Write an algorithm primedivisors(n) to compute the list of divisors of n that are prime numbers. (Hint: Compute divisors(n) and then filter out the primes in this list.)
5.
(*) We can also find the gcd of two numbers by computing prime factorisation of both the numbers. Try to write an algorithm to compute the prime factorisation of a number.
- (i) The prime factorisation of 180 is 2² × 3² × 5¹. How would you represent this?
- (ii) How would you compare the prime factorisations of two numbers?

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