Explore fundamental concepts of Highest Common Factor (HCF) and Least Common Multiple (LCM) through practical, real-life scenarios. This quiz helps reinforce skills in solving everyday problems using HCF and LCM techniques for students and learners seeking to strengthen their mathematics knowledge.
If a seminar is held every 6 days and a workshop every 8 days, after how many days will both occur together again?
Explanation: The correct answer is 24, which is the least common multiple (LCM) of 6 and 8. LCM is used here as we're trying to find when both events coincide. The options 12 and 14 are not multiples of both numbers, and 48 is also a common multiple but not the smallest one, making it incorrect for this scenario.
A tailor has two ribbons of lengths 18 meters and 24 meters. What is the greatest possible length of equal pieces he can cut from both without any ribbon left over?
Explanation: The correct answer is 6, as it is the highest common factor (HCF) of both ribbon lengths. While 12 is a common factor, it is not the greatest; 8 is not a factor of 18, and 3 is smaller than the HCF. Only 6 divides both 18 and 24 exactly without leaving any leftover ribbon.
If a school has 45 blue chairs and 30 red chairs, what is the largest number of identical groups that can be formed so that each group has the same number of blue and red chairs with none left over?
Explanation: 15 is the HCF of 45 and 30, allowing for the largest identical groups. 10 and 5 are common factors but not the greatest, which would leave more groups with fewer chairs. 3 also divides both but creates even smaller groups, so only 15 is correct for maximizing group size.
A red light blinks every 9 seconds and a green light blinks every 12 seconds. After how many seconds do both lights blink together again?
Explanation: 36 is the least common multiple of 9 and 12, so both lights will again blink together after 36 seconds. 18 and 21 are multiples of one or the other, not both. 48 is a common multiple but not the smallest, so it is less appropriate for this case.
You have 20 lemon candies and 32 orange candies. What is the largest number of identical candy packets you can make with no candies left over in any packet?
Explanation: 4 is the highest common factor of 20 and 32, making it possible to pack candies with none left over. 8 and 16 do not divide both numbers exactly; 12 is not a common factor. Only 4 divides both 20 and 32 evenly, making it the largest possible packet number.
A gardener waters plants every 15 days and trims hedges every 20 days. In how many days will both tasks fall on the same day again?
Explanation: The correct answer is 60, which is the LCM of 15 and 20. Only at 60 days will both tasks coincide again. 30 and 45 are not multiples of both numbers, and although 35 is a number in the list, it does not fit. LCM helps solve synchronization problems like this.
If you have 48 milk chocolates and 36 dark chocolates, how many gift boxes with an equal number of each chocolate can you prepare with none left over?
Explanation: The highest common factor (HCF) between 48 and 36 is 12, so you can make 12 identical boxes with equal chocolates of each type. The options 8, 9, and 6 are factors, but not the highest that fits both numbers, therefore, they would not maximize box numbers.
Flower pots are available in packs of 9 and packs of 12. What is the minimum number of pots you need to buy if you want to have the same number of pots from both types and none left unused?
Explanation: The minimum number is 36, as it is the LCM of 9 and 12, meaning you can buy 4 packs of 9 and 3 packs of 12 for equal numbers. 27 is a multiple of 9 but not 12, and 72 and 108 are higher multiples, but not the smallest possible.
Two athletes start from the same point and run around a circular track at intervals of 20 and 30 minutes per lap, respectively. After how many minutes will both meet at the starting point?
Explanation: 60 is the least common multiple of 20 and 30, which means both athletes will be at the starting point together after 60 minutes. 50 and 40 are not multiples of both intervals, and 90 is a common multiple but comes after 60. LCM is used for such timing problems.
A teacher has 36 pencils and 54 erasers. What is the greatest number of students that can receive an equal set of pencil and eraser with none left?
Explanation: 18 is the HCF of 36 and 54, allowing the greatest possible number of students to receive equal sets without leftovers. 12, 9, and 6 do divide both numbers but result in fewer students than maximum. HCF provides the solution for distributing items equally.