Question 1: Common factors are numbers that divide two or more numbers exactly without leaving a remainder. Recognizing the greatest common factor (GCF) allows us to simplify fractions efficiently.

Answer:

Question 1: Find the greatest common factor (GCF) of 48 and 60.

Answer: 12

Question 2: Simplify the fraction 36/54 by dividing numerator and denominator by their GCF.

Answer: 2/3

Question 3: Determine the GCF of 81 and 54.

Answer: 27

Question 4: Calculate the GCF of 100, 150, and 200.

Answer: 50

Question 1: A recipe calls for 120 grams of sugar and 180 grams of flour. Find the largest amount of each ingredient that can be used while maintaining the same ratio, by dividing both by their GCF. Explain why this simplifies the ingredients.

Answer: The GCF is 60. Dividing both by 60 gives 2 parts sugar and 3 parts flour, maintaining the original ratio.

Question 2: Two numbers have a GCF of 12. If one number is 36, what could the other number be? Justify your answer.

Answer: The other number must be a multiple of 12; for example, 24, 48, or 60. Since 36's factors include 12, the other number must share 12 as a factor.

Question 3: A school has 84 students and 126 books to be distributed evenly among classroom groups. Find the maximum number of students per group and the maximum number of books per group. Explain your reasoning.

Answer: GCF of 84 and 126 is 42. So, maximum students per group is 42, and maximum books per group is 42, maintaining even distribution.

Question 1: A factory produces 1200 widgets and 1500 gadgets daily. Find the largest number of identical boxes they can use to pack each product without mixing them, based on common factors. How many items will be in each box?

Answer: GCF of 1200 and 1500 is 300. So, they can pack 300 items per box: 4 boxes of widgets and 5 boxes of gadgets.

Question 1: Find two numbers greater than 50 that have a GCF of 25. Provide one example and justify your choice.

Answer: Example: 75 and 100. Both are divisible by 25, and their GCF is 25.

Question 2: Construct two numbers whose GCF is 18 and LCM is 288. Show your calculations and reasoning.

Answer: Prime factorization: 18 = 2×3²; 288 = 2⁵×3². Possible numbers: 36 and 144, because GCF is 18 and LCM is 288.

Question 1: Find the common factors of 56, 72, and 84.

Answer: 1, 2, 4

Question 2: Simplify the fraction 45/75 using the GCF.

Answer: 3/5

Question 3: Identify the common factors that are shared by 18 and 24, but not by 36.

Answer: 1 and 2

Question 1: A student finds the GCF of 36 and 48 as 12, but then they divide 36/12 and 48/12 to simplify the fractions. Identify the mistake in this process.

Answer: The mistake is dividing both numerator and denominator separately when simplifying fractions; they should divide both by their GCF to correctly simplify the fraction.

Question 2: A student claims that the GCF of 50 and 75 is 25, but they don't check whether it is the greatest. Is this correct? Explain.

Answer: Yes, 25 is the GCF because it is the largest number dividing both 50 and 75 exactly.