Prime Factorization Lcm Gcf Prime Composite Irrational Rational

Prime factorization of 31

• 1, 31

• 31, 62, 93

• 31

• 25 + 6

What is the prime factorization of 30?

• 1 x 2 x 3 x 5

• 1, 2, 3, 5, 6, 10, 15, 30

• 30, 60, 90

• 3

Rachel and Makayla are masters at GCF and LCM using Venn diagrams. However, they get confused when they calculate the GCF and LCM on two numbers. Rachel thinks that to determine which answer is the GCF she remembers that the “G” stands for greatest so the answer that is greater is the GCF and the “L” stands for least. She also learned this while teaching in Azerbaijan. Makayla says not so fast. Makayla thinks the LCM is the greater answer.

• Rachel is correct. The GCF answer is greater because “G” stands for greatest and “L” stands for least. Also, the F” stands for factors and the “M” stands for multiples. The factors of two numbers are always greater than or equal to the number and multiples are always less than or equal to the number.

• Rachel is not correct. It is the opposite of what you would think because the “F” stands for factors and the “M” stands for multiples. The factors of two numbers are always less than or equal to the number and multiples are always greater than or equal to the number.

• Makayla cannot be correct because “L” stands for least!

• Makayla is correct because the “C” stands for common in both GCF and LCM. When Makayla compared the common factors of two numbers to the common multiples of the same two numbers she discovered that none of the numbers were the same.

Cassie and Mike are arguing and ask for your help. Cassie thinks relatively prime numbers are only two prime numbers that are relatives.

• Cassie is almost correct. Only one out of the two numbers need to be prime in order for them to be relatively prime. For example, GCF (3, 25) = 1

• Mike argues that this cannot be correct because relatively prime numbers are not only prime numbers, they can be two natural numbers too. He states GCF (16, 25) = 1

• Cassie proves it by stating GCF (3, 5) = 1 and 3 and 5 are like prime relatives with basically nothing in common but 1 as a factor.

• Mike argues that only two “odd” numbers can be relatively prime with this example, GCF (9, 15) = 1

Meghan loves conducting math research on the Internet. On her trip to Spain, she discovered that this guy named Gauss figured out in the 1800’s that every integer greater than 1 is a prime number or a product of prime numbers. She is wondering if two of these integers could have the same prime factorization.

• No, because each integer except 1 has a unique prime factorization and does not include a composite number in the answer.

• No, due to the fact that each integer except 1 has more than one way to represent their product of prime numbers.

• Yes, they could because the prime factorizazation for each integer is not unique and can include composite numbers in the answer.

• Yes, they could because 8 and 16 have products that only have the prime number 2!

Mackenzie and Joanna love creating natural numbers with exactly 6 factors because they are going to be math teachers some day. Mackenzie created the number 6 with an exponent of 5. Joanna created the number 7 with an exponent of 5.

• Only Joanna’s is correct because the base needs to be a prime number.

• Only Mackenzie’s is correct because the base needs to be a composite number based on the rectangular array theorem.

• Mackenzie and Joanna are both correct since they both have an exponent of 5 and the number of factors is the exponent plus 1.

• They are both wrong.

Grace enjoys discovering mathematical concepts through patterns. She discovered based on her organized table of data that as the number 2 is raised to greater and greater exponents that the number of factors only increase by 1 more factor each time the exponent grows by 1.

• Grace is on the right track but her table is designed with information missing to reach full understanding.

• Grace should not have started her table with looking at the number two because it is a prime number and it should be a composite number to prove her theory.

• Grace has not had enough sleep over the last week and is getting confused.

• Grace is correct with her theory.

Mrs. J has students that are easily tricked while looking at the Real Number diagram. She questions them on whether 2.45382... is a rational number or an irrational number.

• Mike jumps up and declares it is irrational because the decimal does not repeat or terminate, like the terminator!

• Cassie gives Mike the stanky face and nicely tells him he is off base. The decimal is rational because all decimals are rational.

• Joanna and Mackenzie always pay attention to Mrs. J and they remember the game they played called “Number Dinner” and tell the class that the decimal is not rational or irrational.

• Griffin says that in baseball you use decimals all the time and are very common so it must be a rational number.

Bailey and Stephen are in a heated debate over whether zero and 1 are either prime or composite. Bailey deep down inside thinks zero is just a bunch of nothingness and wants to go watch Netflix. Stephen just wants to stop debating and go to Starbucks to relax.

• No way are they composite because they do not have more than 2 factors. So, they must be prime numbers.

• No way, neither, they are not greater than 1 and not a product of primes.

• They are definitely neither because they each have one as a factor.

• A prime number has exactly 2 distinct factors, so they are composite.

Ambur P made a conjecture that if you multiply 6 raised to the exponent 5 and 7 raised to the exponent 5 that you will have 100 factors. Grace just rolled into class with her hair dripping wet from swim practice and states, there is no way this can be correct.

• Ambur P is correct because 5 is prime.

• Ambur P is correct because you add one to each exponent and then multiply.

• Grace is correct because 5 is prime.

• Grace is correct because 6 is not prime.