Hard Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Write the prime factorization of 210. 210 = × × ×

(2) Write the HCF of 48 and 180 using the prime factorization method. HCF =

Perfect! HCF takes the lowest powers of common prime factors.

(3) Write the LCM of 18 and 24. LCM =

Excellent! LCM takes the highest powers of all prime factors.

(4) State the condition for a number to be divisible by 4.

A number is divisible by 4 if its lastsum digits form a number by .

(5) Write one example of an irrational number. 2227

Correct! Examples include 2, π, 3, 5, etc.

Short Answer Questions (2 Marks Each)

Answer each question with proper working

(1) Find the HCF of 135 and 225 using Euclid's division lemma. HCF =

Perfect! When remainder becomes 0, the divisor is the HCF.

(2) Express 540 as the product of prime factors. 540 = 2223 × 3332 ×

(3) Without actually performing the long division, find the remainder when x² + 2x + 3 is divided by x - 2 using the Remainder Theorem. Remainder =

Excellent! Remainder = 11.

(4) Find the LCM and HCF of 72 and 90 and verify the relation HCF × LCM = product of numbers. HCF = and LCM =

Verification: 18 × 360 = 6480 = 72 × 90 ✓

(5) Show that 72 is irrational.

Long Answer Questions (4 Marks Each)

(1) Using Euclid's division lemma, show that any positive odd integer is of the form 4m + 1 or 4m + 3, where m is a non-negative integer.

(2) Find the HCF of 867 and 255 using Euclid's algorithm. HCF =

Perfect! HCF of 867 and 255 = 51.

(3) Three different containers contain 496 litres, 403 litres, and 713 litres of milk respectively. Find the maximum capacity of a container that can measure the milk of each container in exact number of times. HCF =

Excellent! Maximum capacity = 31 litres.

(4) Show that 5 + 7 is an irrational number.

(5) The traffic lights at three different road crossings change after every 48 seconds, 72 seconds, and 108 seconds respectively. If they all change simultaneously at 7 a.m., at what time will they change together again? LCM = seconds which gives the Time: 7 : 07 : 127 : 07 : 367 : 06 : 517 : 05 : 12 a.m.

Next time together: 7 : 00 : 00 a.m. + 7 min 12 sec = 7 : 07 : 12

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) The prime factorization of 210 is:

(a) 2 × 3 × 5 × 7 (b) 2 × 3 × 35 (c) 22 × 3 × 5 × 7 (d) 2 × 32 × 5 × 7

Correct! 210 = 2 × 3 × 5 × 7.

(2) The HCF of 48 and 180 is:

(a) 6 (b) 12 (c) 24 (d) 18

Correct! 48 = 24 × 3, 180 = 22 × 32 × 5, so HCF = 22 × 3 = 12.

(3) The LCM of 18 and 24 is:

(a) 72 (b) 36 (c) 48 (d) 60

Correct! 18 = 2 × 32, 24 = 23 × 3, so LCM = 23 × 32 = 72.

(4) Which of these is NOT a perfect square?

(a) 144 (b) 196 (c) 225 (d) 210

Correct! 144 = 122, 196 = 142, 225 = 152, but 210 is not a perfect square.

(5) The smallest number divisible by both 8 and 12 is:

(a) 96 (b) 48 (c) 24 (d) 12

Correct! LCM of 8 and 12 = 24.

(6) Which of the following is irrational?

(a) 45 (b) 0.333... (c) 3 (d) 0.25

Correct! 3 is irrational because 3 is not a perfect square.

(7) The remainder when 145 is divided by 12 is:

(a) 0 (b) 1 (c) 2 (d) 3

Correct! 145 = 12 × 12 + 1, so remainder = 1.

(8) The least number that is divisible by 12, 18, and 24 is:

(a) 72 (b) 144 (c) 48 (d) 36

Correct! LCM of 12, 18, and 24 = 72.

(9) If HCF of two numbers is 18 and their product is 1296, then their LCM is:

(a) 72 (b) 36 (c) 48 (d) 54

Correct! Using HCF × LCM = Product: 18 × LCM = 1296, so LCM = 72.

(10) The LCM of 48, 72, and 108 is:

(a) 432 (b) 864 (c) 216 (d) 144

Correct! LCM = 24 × 33 = 432.

Complex Real Number Properties True or False

Determine whether these statements are True or False:

Real Numbers - Hard Quiz

🎉 Outstanding Mastery! Advanced Real Number Excellence Achieved:

You have successfully conquered the "Real Numbers (Hard)" worksheet and mastered:

(1) Advanced Prime Factorization: Factorizing large composite numbers like 210, 540 into their complete prime factor forms

(2) Complex HCF Calculations: Using Euclid's algorithm for challenging number pairs like 867 and 255

(3) Multi-number LCM: Finding LCM for three numbers simultaneously using systematic prime factorization

(4) Advanced Irrationality Proofs: Proving irrationality of complex expressions like √5 + √7 and 7√2

(5) Divisibility Rule Mastery: Understanding and applying advanced divisibility conditions

(6) Euclidean Algorithm Expertise: Implementing Euclid's division lemma for complex calculations

(7) Mathematical Proof Techniques: Using proof by contradiction and direct proof methods

(8) Remainder Theorem Applications: Finding polynomial remainders without long division

(9) Form Analysis: Proving that odd integers take specific forms like 4m+1 or 4m+3

(10) Complex Real-world Problems: Solving practical scenarios involving multiple constraints

(11) Time Synchronization Problems: Using LCM to solve traffic light and timing problems

(12) Advanced Relationship Verification: Confirming HCF × LCM = Product relationships

(13) Three-number HCF Problems: Finding HCF of three numbers using systematic approaches

(14) Perfect Square Recognition: Identifying and distinguishing perfect squares from other numbers

(15) Advanced Number Classification: Categorizing numbers as rational, irrational, prime, composite

(16) Computational Accuracy: Maintaining precision in complex multi-step calculations

(17) Mathematical Communication: Expressing complex proofs and solutions clearly

(18) Strategic Problem-solving: Choosing optimal methods for different types of number theory problems

(19) Logical Reasoning: Developing sophisticated mathematical arguments and proofs

Exceptional achievement! You've mastered advanced real number theory with sophisticated mathematical reasoning!