Easy Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Find the HCF of 18 and 24.

(2) Write the prime factorisation of 60.

60 = × × ×

Correct! 60 = 4 × 15 = 4 × 3 × 5 = 22 × 3 × 5.

(3) What is the LCM of 12 and 15?

Perfect! LCM(12, 15) = 22 × 3 × 5 = 60.

(4) Write any one pair of co-prime numbers. 7 and 96 and 84 and 68 and 12

Excellent! 7 and 9 have HCF = 1, so they are co-prime.

(5) Write the smallest positive integer that is divisible by both 4 and 6.

Correct! The smallest positive integer divisible by both is their LCM = 12.

Short Answer Questions (2 Marks Each)

(1) Find the HCF and LCM of 36 and 48 using prime factorisation method. HCF = and LCM =

Perfect! 36 = 22 × 32, 48 = 24 × 3. HCF = 22 × 3 = 12, LCM = 24 × 32 = 144.

(2) Express 98 as a product of its prime factors. 98 = × ×

Excellent! 98 = 2 × 49 = 2 × 7 × 7 = 2 × 72.

(3) If HCF(8, x) = 4 and LCM(8, x) = 48, find the value of x. x =

Perfect! Using HCF × LCM = Product: 4 × 48 = 8 × x, so x = 1928 = 24.

(4) Prove that √3 is irrational.

(5) Find the HCF and LCM of 20, 28 using the division method. HCF = and LCM =

Excellent! Using Euclid's algorithm: HCF = 4, and LCM = 20×284 = 140.

Long Answer Questions (4 Marks Each)

(1) Use Euclid's division algorithm to find the HCF of 135 and 225. HCF =

Perfect! 225 = 135 × 1 + 90, 135 = 90 × 1 + 45, 90 = 45 × 2 + 0. So HCF = 45.

(2) Show that any positive odd integer is of the form 6q + 1, 6q + 3, or 6q + 5, where q is an integer.

(3) Find the HCF and LCM of 24 and 36, and verify that HCF × LCM = Product of the two numbers. HCF = , LCM =

Excellent! HCF × LCM = 12 × 72 = 864 = 24 × 36.

(4) Prove that 2 is irrational.

(5) A and B walk around a circular track. A takes 6 minutes to complete one round, and B takes 8 minutes. After how many minutes will they meet at the starting point? minutes

Perfect! They meet after LCM(6, 8) = 24 minutes.

Part B: Objective Questions (1 Mark Each)

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

(1) The HCF of 16 and 24 is

(a) 4 (b) 8 (c) 12 (d) 16

Correct! 16 = 24, 24 = 23 × 3, so HCF = 23 = 8.

(2) LCM of 5 and 20 is

(a) 25 (b) 100 (c) 20 (d) 15

Correct! Since 20 = 5 × 4, the LCM is 20.

(3) The number 5 is

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

Correct! 5 is irrational since 5 is not a perfect square.

(4) Two numbers are said to be co-prime if

(a) Their sum is 1 (b) Their HCF is 1 (c) Their LCM is 1 (d) They are both even

Correct! Co-prime numbers have no common factors other than 1.

(5) The prime factorisation of 90 is

(a) 2 × 3 × 3 × 5 (b) 2 × 5 × 5 × 3 (c) 2 × 2 × 3 × 5 (d) 3 × 5 × 5

Correct! 90 = 2 × 45 = 2 × 9 × 5 = 2 × 32 × 5.

(6) The product of HCF and LCM of two numbers is

(a) Always equal to their sum

(b) Equal to the product of the numbers

(c) Equal to the larger number

(d) Cannot be determined

Correct! This is a fundamental theorem: HCF(a,b) × LCM(a,b) = a × b.

(7) Euclid's division lemma is used to

(a) Find square roots

(b) Find factors

(c) Find irrational numbers

(d) Find HCF of two numbers

Correct! Euclid's algorithm uses the division lemma to find HCF efficiently.

(8) If two positive integers a and b are such that a = 4b, then HCF(a, b) is

(a) a (b) b (c) 4 (d) 2b

Correct! Since a = 4b, b divides a, so HCF(a,b) = b.

(9) Which of the following is a rational number?

(a) 2 (b) π (c) 57 (d) 3

Correct! 57 is a fraction of integers, so it's rational.

(10) Which of the following is not a prime number?

(a) 2 (b) 3 (c) 9 (d) 5

Correct! 9 = 3 × 3, so it's composite, not prime.

Real Numbers Challenge

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

Real Numbers Quiz