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Class 9 Maths Part 2 Chapter 11 – The World of Algorithms | Ganita Manjari Worksheet

Practise Class 9 Maths Part 2 Chapter 11 – The World of Algorithms with this Ganita Manjari worksheet. Prepare more with important questions and revision exercises designed to strengthen understanding of algorithms, improve mathematical practice and support effective preparation for class tests and examinations.

Class 9 Maths Chapter 11 The World of Algorithms – Fresh Worksheet
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Class 9 Maths – Chapter 11

The World of Algorithms – Fresh Worksheet

Practice Worksheet based on algorithms, divisors, GCD, Euclid’s algorithm, Aryabhata’s division algorithm and prime numbers.

Name: __________________________     Class: __________     Date: __________
Instructions:
  • Read every question carefully.
  • Show the steps of the algorithm wherever required.
  • Do not skip intermediate steps in GCD questions.
  • Try to solve the worksheet independently before checking the solutions.

Section A – Understanding Algorithms

Q1.
What is an algorithm? Write any two characteristics of a good algorithm.
Solution:

An algorithm is a clear, step-by-step procedure used to solve a problem or complete a task.

Two characteristics of a good algorithm are:

  1. Its steps should be clear and precise.
  2. It should eventually produce the required answer.
Q2.
Why is it important to write the steps of an algorithm in the correct order?
Solution:

The steps of an algorithm depend on one another. If we change their order, the required information may not be available at the correct time and the final answer may be wrong.

Q3.
Write an algorithm to add two 3-digit numbers.
Solution:
  1. Write the two numbers one below the other and align their digits from right to left.
  2. Add the digits in the units column.
  3. If the sum is 10 or more, write the units digit and carry 1.
  4. Add the tens digits along with the carry.
  5. Continue in the same way for the hundreds column.
  6. If a carry remains after the last column, write it on the left.

Section B – Divisors

Q4.
Find all the divisors of 24.
Solution:

The numbers that divide 24 exactly are:

1, 2, 3, 4, 6, 8, 12, 24

Q5.
Find all the divisors of 36 and write them in increasing order.
Solution:

The divisors of 36 are:

1, 2, 3, 4, 6, 9, 12, 18, 36

Q6.
Write the divisors of 48 in pairs.
Solution:

The divisor pairs are:

(1, 48), (2, 24), (3, 16), (4, 12), (6, 8)

Section C – Greatest Common Divisor

Q7.
Find gcd(18, 30) by listing the divisors of both numbers.
Solution:

Divisors of 18:

1, 2, 3, 6, 9, 18

Divisors of 30:

1, 2, 3, 5, 6, 10, 15, 30

Common divisors:

1, 2, 3, 6

Therefore, gcd(18, 30) = 6.

Q8.
Find gcd(42, 70) using the divisor method.
Solution:

Divisors of 42:

1, 2, 3, 6, 7, 14, 21, 42

Divisors of 70:

1, 2, 5, 7, 10, 14, 35, 70

Common divisors:

1, 2, 7, 14

Therefore, gcd(42, 70) = 14.

Q9.
Find gcd(84, 126) using Euclid’s subtraction algorithm.
Solution:

Start with:

gcd(126, 84)

= gcd(84, 126 − 84)

= gcd(84, 42)

= gcd(42, 42)

= 42

Therefore, gcd(84, 126) = 42.

Section D – Aryabhata’s Division Algorithm

Q10.
Find gcd(48, 18) using Aryabhata’s division algorithm.
Solution:

48 = 18 × 2 + 12

Therefore, gcd(48,18) = gcd(18,12)

18 = 12 × 1 + 6

Therefore, gcd(18,12) = gcd(12,6)

12 = 6 × 2 + 0

Therefore, gcd(48,18) = 6.

Q11.
Find gcd(135, 45) using the division algorithm.
Solution:

135 = 45 × 3 + 0

The remainder is 0, so the divisor 45 is the GCD.

gcd(135,45) = 45

Q12.
Find gcd(252, 105) using Aryabhata’s division algorithm.
Solution:

252 = 105 × 2 + 42

105 = 42 × 2 + 21

42 = 21 × 2 + 0

The last non-zero remainder is 21.

Therefore, gcd(252,105) = 21.

Section E – Prime Numbers

Q13.
Write all the prime numbers between 10 and 30.
Solution:

The prime numbers between 10 and 30 are:

11, 13, 17, 19, 23, 29

Q14.
Write an algorithm prime(n) to check whether n is a prime number.
Solution:
  1. If n is less than 2, report that n is not prime.
  2. Find all the divisors of n.
  3. If n has exactly two divisors, 1 and n, report that n is prime.
  4. Otherwise, report that n is not prime.
Q15.
Is 37 a prime number? Use its divisors to justify your answer.
Solution:

The divisors of 37 are:

1 and 37

It has exactly two divisors. Therefore, 37 is a prime number.

Section F – Prime Divisors and Factorisation

Q16.
Find the prime divisors of 60.
Solution:

Divisors of 60 include:

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

The prime divisors are:

2, 3 and 5

Q17.
Find the prime factorisation of 120.
Solution:

Divide 120 repeatedly by prime numbers:

120 = 2 × 60

= 2 × 2 × 30

= 2 × 2 × 2 × 15

= 2 × 2 × 2 × 3 × 5

Therefore, 120 = 23 × 3 × 5.

Q18.
Find the prime factorisation of 180.
Solution:

180 = 2 × 90

= 2 × 2 × 45

= 2 × 2 × 3 × 15

= 2 × 2 × 3 × 3 × 5

Therefore, 180 = 22 × 32 × 5.

Section G – Thinking and Reasoning

Q19.
A student says that 1 is a prime number because it has only one divisor. Is the student correct? Explain.
Solution:

No, the student is not correct.

A prime number must have exactly two distinct divisors: 1 and the number itself.

The number 1 has only one divisor, which is 1. Therefore, 1 is neither prime nor composite.

Q20.
Without listing all the divisors, find gcd(96, 36) using the division algorithm.
Solution:

96 = 36 × 2 + 24

36 = 24 × 1 + 12

24 = 12 × 2 + 0

The last non-zero remainder is 12.

Therefore, gcd(96,36) = 12.

Q21.
Two numbers have prime factorisations 23 × 3 × 5 and 22 × 32 × 7. What are their common prime factors?
Solution:

The first number has prime factors 2, 3 and 5.

The second number has prime factors 2, 3 and 7.

The prime factors common to both numbers are:

2 and 3

Q22.
Explain why the division algorithm can be much faster than checking every number from 1 up to n to find the GCD.
Solution:

The divisor method may require checking many numbers one by one. For large numbers, this can take a lot of time.

The division algorithm uses remainders to make the numbers smaller at every step. Therefore, the number of steps is much smaller.

Hence, the division algorithm is much more efficient for finding GCDs of large numbers.

Challenge Questions

Q23.
Find gcd(693, 315) using Aryabhata’s division algorithm.
Solution:

693 = 315 × 2 + 63

315 = 63 × 5 + 0

Therefore, gcd(693,315) = 63.

Q24.
Write an algorithm to find the greatest common divisor of two positive integers using the division method.
Solution:
  1. Let the two numbers be m and n.
  2. If m is smaller than n, interchange m and n.
  3. Find the remainder when m is divided by n.
  4. If the remainder is 0, report n as the GCD.
  5. Otherwise, replace m by n and n by the remainder.
  6. Repeat the process until the remainder becomes 0.
  7. The last non-zero divisor is the GCD.
Q25.
Design an algorithm to find all the prime divisors of a number n.
Solution:
  1. Find all the divisors of n.
  2. Take each divisor one at a time.
  3. Check whether the divisor is prime.
  4. If it is prime, add it to the list of prime divisors.
  5. Continue until all divisors have been checked.
  6. Report the final list.
Quick Revision:
  • An algorithm is a step-by-step method for solving a problem.
  • A divisor divides a number exactly without leaving a remainder.
  • The GCD is the greatest number that divides both numbers.
  • Euclid’s division algorithm repeatedly uses remainders.
  • A prime number has exactly two distinct divisors: 1 and itself.
  • Prime factorisation expresses a number as a product of prime numbers.

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