Hard Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Define a rational number. Give an example. A rational number can be expressed as where p and q are integers and q ≠ .

Perfect! Rational numbers have the form pq where q is non-zero.

(2) State Euclid's division lemma. For any two positive integers a and b, there exist uniquecommon integers q and r such that a = + r, where q, r are the quotient and remainder respectively, and 0 ≤ r < .

Excellent! This is the foundation of the division algorithm.

(3) Can the square root of 2 be expressed as a rational number? Justify. NoYes

Because if 2 = pq, then 2q2 = p2, making both p and q are , contradicting that pq is in terms.

Perfect! This is a proof by contradiction showing 2 is irrational.

(4) Write the HCF of two consecutive even numbers. HCF =

Correct! Consecutive even numbers differ by 2, so their HCF is 2.

(5) Is 0 a rational number? Explain. YesNo

Excellent! Zero satisfies the definition of a rational number as 0 can be written as 01 where 1 ≠ 0.

Short Answer Questions (2 Marks Each)

(1) Use Euclid's division algorithm to find the HCF of 960 and 432.

960 = × 23 +

432 = 96 × +

96 = 48 × +

Therefore, HCF(960, 432) =

Excellent systematic application of Euclid's algorithm!

(2) Show that any positive even integer is of the form 2q, where q is an integer.

Let n be any positive even integer. By definition, n is divisible by

Therefore, n = 2 × for some integer q ≥

Perfect! This shows the general form of even numbers.

(3) Find the LCM and HCF of 65 and 117 and verify that HCF × LCM = Product of the two numbers.

Using Euclid's algorithm: HCF(65, 117) =

LCM(65, 117) =

Verification: HCF × LCM = 13 × 585 = = 65 × 117

Excellent verification of the fundamental relationship!

(4) If the HCF of 210 and 55 is expressible in the form 210x + 55y, find the values of x and y. x = , y =

Excellent application of the extended Euclidean algorithm!

(5) Prove that 3+25 is irrational.

Long Answer Questions (4 Marks Each)

(1) Show that every positive integer is either a prime or a product of primes.

(2) Use Euclid's algorithm to find the HCF of 867 and 255. Express it as a linear combination.

867 = × 37 +

255 = 102 × +

102 = 51 × +

HCF =

Working backwards: 51 = 255 - 102 × 2 = 255 - (867 - 255 × 3) × 2

51 = 255 - 867 × 2 + 255 × 6 = 255 × 7 - 867 × 2

Therefore: 51 = 867 × () + 255 × ()

Perfect application of extended Euclidean algorithm!

(3) Prove that there are infinitely many prime numbers.

(4) The decimal expansion of a rational number is 0.245245245... Express it in the form pq.

(5) A P.T. instructor wants to arrange 120 students in rows such that each row has the same number of students. What is the maximum number of students that can be arranged in a row? students per row

Perfect application of factors in real-world context!

Part B: Objective Questions (1 Mark Each)

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

(1) The HCF of 867 and 255 is

(a) 51 (b) 3 (c) 15 (d) 255

Correct! Using Euclid's algorithm: 867 = 255×3 + 102, 255 = 102×2 + 51, 102 = 51×2 + 0.

(2) The decimal expansion of a rational number is always

(a) Non-repeating and non-terminating (b) Repeating or terminating (c) Irrational (d) Finite only

Correct! Rational numbers always have terminating or repeating decimal expansions.

(3) If a number is of the form 6q + 5, it can be

(a) Even only (b) Odd only (c) Prime only (d) Composite only

Correct! 6q is always even, so 6q + 5 is always odd.

(4) The Fundamental Theorem of Arithmetic states that

(a) Every even number is divisible by 2

(b) Every number is a multiple of its factors

(c) Every composite number can be expressed as a product of primes uniquely

(d) Prime numbers cannot be factored

Correct! This theorem guarantees unique prime factorization (up to order).

(5) 2 is

(a) Rational (b) Integer (c) Irrational (d) Natural number

Correct! 2 cannot be expressed as pq where p, q are integers.

(6) If a and b are two integers such that a = bq + r, then r must satisfy

(a) r > b (b) 0 ≤ r < b (c) r < 0 (d) r ≥ b

Correct! This is the constraint in Euclid's division lemma.

(7) The LCM of two numbers is 180 and their HCF is 6. If one number is 30, the other is

(a) 36 (b) 12 (c) 24 (d) 45

Correct! Using HCF × LCM = product of numbers: 6 × 180 = 30 × x, so x = 36.

(8) Which of the following numbers has a non-terminating, repeating decimal expansion?

(a) 13 (b) 78 (c) 52 (d) 34

Correct! 13 = 0.333... (repeating), while others terminate.

(9) What is the smallest number which when divided by 12, 15, and 20 leaves a remainder of 3 in each case?

(a) 60 (b) 63 (c) 123 (d) 243

Correct! LCM(12,15,20) = 60. Number = 60k + 3, smallest is 63.

(10) If x and y are positive integers such that LCM(x, y) = 120 and HCF(x, y) = 6, then x·y =

(a) 720 (b) 126 (c) 90 (d) 102

Correct! Using the relationship: HCF × LCM = x × y, so 6 × 120 = 720.

Real Numbers Challenge

Determine whether these statements about real numbers are True or False:

Real Numbers Quiz

🎉 You Did It! What You've Learned:

By completing this worksheet, you now have a solid understanding of:

(1) Number Classification: Distinguishing between rational and irrational numbers

(2) Euclid's Algorithm: Finding HCF using systematic division process

(3) Fundamental Theorem of Arithmetic: Understanding unique prime factorization

(4) LCM and HCF Relationship: Using HCF × LCM = Product of numbers

(5) Decimal Expansions: Converting between fractions and decimal forms

(6) Proof Techniques: Using contradiction to prove irrationality

(7) Linear Combinations: Expressing HCF as ax + by using extended Euclidean algorithm

(8) Real-world Applications: Solving practical problems using number theory concepts

Excellent work mastering the advanced concepts of real numbers and their properties!