Practice Questions
Finding Common Ground
Define the term 'Highest Common Factor' (HCF).
Design a real-world scenario where finding the LCM of 15, 20, and 30 would be necessary to solve a problem related to scheduling.
A student states that the LCM of any two co-prime numbers is always 1. Critique this statement.
A student claims that the HCF of any two distinct prime numbers is always 1. Justify this claim.
Design a word problem involving three different quantities where the solution requires finding the HCF of three numbers, each greater than 50.
Define the term 'Lowest Common Multiple' (LCM).
Calculate the Highest Common Factor (HCF) of 18 and 27 using the prime factorization method.
Calculate the Lowest Common Multiple (LCM) of 20 and 30.
Two numbers have an HCF of 6 and an LCM of 72. If one of the numbers is 18, apply the relationship between HCF, LCM, and the product of two numbers to find the other number.
What is meant by 'prime factorization' of a number?
List the first five multiples of the number 7.
Name the only even prime number.
List all the factors of the number 48.
Two wires are 12 m and 16 m long. The wires are to be cut into pieces of equal length without any waste. Calculate the greatest possible length of each piece.
Identify the prime factorization of 100 using the division method.
Design a step-by-step procedure to find the smallest number that when divided by 12, 15, and 20 leaves a remainder of 5 in each case. Justify each step of your procedure and find the number.
Calculate the HCF and LCM of 90 and 144.
Analyze the numbers 15 and 28. Are they co-prime? Calculate their HCF to justify your answer.
Formulate a general rule for the LCM of a number '' and its multiple '', where '' is a positive integer. Justify your rule.
Explain what co-prime numbers are. Provide an example of a pair of co-prime numbers.
List the factors of 18 and 30. Then, identify their common factors.
List the first six multiples of 6 and 9. Then, identify their first two common multiples.
Describe the process of finding the HCF of two numbers by listing their factors. Use the numbers 28 and 42 to illustrate your description.
Describe the process of finding the LCM of two numbers by listing their multiples. Use the numbers 8 and 12 to illustrate your description.
A florist has 48 roses and 60 lilies. She wants to make identical bouquets with both types of flowers, using all the flowers. Analyze the situation to find the greatest number of identical bouquets she can make.
Find the prime factorization of the number 420.
Three traffic lights at different crossings change after every 48 seconds, 72 seconds, and 108 seconds, respectively. If they all change simultaneously at 8:00 AM, at what time will they again change simultaneously? Analyze the problem to find the solution.
Demonstrate that the product of two consecutive even numbers is always divisible by 8, by taking the example of 10 and 12.
Two tankers contain 850 litres and 680 litres of kerosene oil respectively. Find the maximum capacity of a container which can measure the kerosene oil of both the tankers when used an exact number of times.
The length and breadth of a rectangular field are 75 m and 60 m respectively. Square tiles are to be laid on the floor. Analyze the dimensions to find the length of the largest possible tile that can be used without cutting.
Evaluate the following statement: If the HCF of two numbers is one of the numbers, then one number must be a multiple of the other. Justify your conclusion.
A student, Rohan, tries to find the HCF of 72 and 108. He writes and . He concludes the HCF is 9 because it's a common factor. Critique Rohan's method and justify the correct procedure to find the HCF.
Two lighthouses flash their lights every 25 seconds and 40 seconds, respectively. If they flash together at 7:00:00 PM, a student calculates they will next flash together at 7:03:00 PM. Evaluate this calculation and justify your answer by finding the correct time.
Create a problem about three runners on a circular track. The runners complete one lap in different times. The problem must require finding the LCM of three numbers to determine when they will all meet at the starting point again for the first time. Assign realistic lap times and solve the problem you created.
Summarize the rule for finding the HCF of two or more numbers using their prime factorizations.
Justify why the product of two numbers is always a multiple of their LCM. Use the relationship HCF × LCM = Product of numbers to support your argument.
Create a pair of composite numbers whose HCF is a prime number greater than 10.
Evaluate the statement: "If the HCF of two numbers is 'h', then the numbers can be expressed as 'hx' and 'hy' where x and y are co-prime." Justify this statement. Then, using this formulation, prove that HCF(a, b) × LCM(a, b) = a × b.
Explain the property that relates the product of two numbers to their HCF and LCM. Show that this property is true for the numbers 15 and 20.
Find the smallest 4-digit number which is exactly divisible by 12, 18, and 24.
A math teacher makes a conjecture: "For any three positive integers a, b, and c, the property HCF(a,b,c) × LCM(a,b,c) = a × b × c holds true." Evaluate this conjecture. Provide a counterexample to disprove it and justify why the property generally only works for two numbers.
Find the smallest number which when divided by 6, 8, and 12 leaves a remainder of 3 in each case.
Formulate algebraic expressions for two consecutive even integers. Propose a value for their HCF and justify why it must always be that value.
In a morning walk, three persons step off together. Their steps measure 80 cm, 85 cm, and 90 cm respectively. Calculate the minimum distance each should walk so that all can cover the same distance in complete steps.
Summarize the rule for finding the LCM of two or more numbers using their prime factorizations.