Real Numbers quiz

The prime factorisation of 156 is:

The HCF of 26 and 91 is:

If HCF(a, b) = 4 and a x b = 960, then LCM(a, b) is:

Which of the following is an irrational number?

The LCM of 8, 9 and 25 is:

The HCF of 2^3 x 3^2 x 5 and 2^2 x 3^3 x 7 is:

The least number that is divisible by all the numbers from 1 to 10 is:

The product of a non-zero rational number and an irrational number is:

If two positive integers p and q are written as p = a^2 b^3 and q = a^3 b, where a and b are primes, then HCF(p, q) is:

The number of prime factors (counting repetitions) of 2^4 x 3^2 x 7 is:

Express 140 and 156 as products of prime factors. (2 marks)

Find the HCF and LCM of 17, 23 and 29. (2 marks)

Prove that 1/sqrt(2) is irrational. (2 marks)

Can two numbers have 18 as their HCF and 380 as their LCM? Give a reason. (2 marks)

Find the HCF of 96 and 404 by prime factorisation and hence find their LCM. (2 marks)

Prove that 7sqrt(5) is irrational. (2 marks)

The HCF of two numbers is 23 and their LCM is 1449. If one number is 161, find the other. (2 marks)

Explain why 3 x 5 x 7 + 7 is a composite number. (2 marks)

Find the LCM and HCF of 26 and 91, and verify that LCM x HCF = product of the numbers. (2 marks)

Prove that 6 + sqrt(2) is irrational. (2 marks)

An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march? (3 marks)

Three traffic lights at different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 a.m., at what time will they next change simultaneously? (3 marks)

A sweet seller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number of barfis and they take up the least area of the tray. What is the number of barfis that can be placed in each stack, and how many stacks are formed? (3 marks)

The floor of a room measures 15 m 17 cm by 9 m 2 cm. Find the side of the largest square tile that can be used to cover the floor exactly, and the number of such tiles required. (3 marks)

Two tankers contain 850 litres and 680 litres of kerosene. Find the maximum capacity of a container that can measure the kerosene of both tankers an exact number of times, and the number of times it is used for each tanker. (3 marks)

Read the passage and answer the questions. A school is organising a fun run. Students are to be arranged in rows with the same number of students in each row and only students of one class in each row. Class IX has 112 students and Class X has 84 students. (i) Write the prime factorisations of 112 and 84. (ii) Find the maximum number of students in each row. (iii) How many rows will be formed in all? (5 marks)

Read the passage and answer the questions. Two bells ring together at 9 a.m. One rings every 15 minutes and the other every 20 minutes. The school uses these bells to signal class changes and breaks. (i) Find the prime factorisations of 15 and 20. (ii) After how many minutes will the bells ring together again? (iii) Find the times at which they will ring together between 9 a.m. and 12 noon. (5 marks)

Read the passage and answer the questions. A teacher writes the number 2^3 x 3 x 5^2 x 7 on the board as the prime factorisation of a number N, and asks the class some questions. (i) Find the value of N. (ii) Does N end with 0? Explain using its prime factors. (iii) Is N divisible by 14 and by 9? Justify using the factorisation. (5 marks)

Read the passage and answer the questions. A student tries to show that sqrt(2) is irrational. She assumes sqrt(2) = a/b, where a and b are coprime integers and b is not 0, and squares both sides to get 2b^2 = a^2. (i) What does 2b^2 = a^2 tell us about a? Which theorem is used? (ii) Writing a = 2c, show that 2 divides b. (iii) Why does this lead to a contradiction, and what is the conclusion? (5 marks)

Read the passage and answer the questions. A florist has 72 roses, 108 lilies and 144 marigolds. He wants to make identical bouquets with the same number of each type of flower in every bouquet, using all the flowers. (i) Find the largest number of bouquets he can make. (ii) How many roses, lilies and marigolds will each bouquet have? (iii) Write the prime factorisations used in your answer. (5 marks)

State the Fundamental Theorem of Arithmetic and explain the meaning of the word "unique" in it. Draw factor trees for 32760 and 7429, write their prime factorisations, and find the HCF and LCM of 336 and 54, verifying HCF x LCM = product. (6 marks)

Prove that sqrt(3) is irrational. Hence prove that 5 - 2sqrt(3) is also irrational. Explain why the method fails if we try to prove sqrt(4) irrational in the same way. (6 marks)

Prove that sqrt(5) is irrational. Show on a number line how a length of sqrt(5) units can be marked using a right triangle with legs 2 and 1 units, and draw the figure. (6 marks)

Find the HCF and LCM of 6, 72 and 120 using the prime factorisation method. Show with an example that the product of the HCF and LCM of three numbers is not, in general, equal to the product of the numbers. Draw factor trees for all three numbers. (6 marks)

Three boys step off together from the same spot. Their steps measure 63 cm, 70 cm and 77 cm respectively. Find the minimum distance each should cover so that all can cover the distance in complete steps. Draw a number line sketch showing the first few steps of each boy, and find the number of steps each takes to cover this distance. (6 marks)