Number Theory - Just be RATIONAL!

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

Tags

CCSS.6.NS.B.4

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!

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.

Factor trees are Kalie’s favorite thing to create in math class because she loves to go to the farmers market. Her friend Claire creates her trees different because she was in Girl Scouts. She leaves her fruit left hanging higher on the tree. They are confused with who is correct.

• Claire is correct because her trees look like the factor trees on Calculator Soup.

• Kalie is correct because each of her rows in her tree equal her original number.

• Both girls need to create their factor trees exactly like their math teacher which is in the shape of a sifter in order to get their answers correct.

• They are both correct but just organize their mathematical thoughts a little differently.

Tags

CCSS.4.OA.B.4

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.

Tags

CCSS.6.EE.A.1

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.

Tags

CCSS.6.EE.A.1

Sara M and Sarah R are usually landing on opposite sides of a discussion because one likes bowling and one chemistry. Definitely two topics that have nothing in common. Sara M stated that if a number is divisible by 8 then it has to be divisible by 2. Sarah R, as predictable, had a theory that was written backwards. This statement would be the converse. She stated that if a number is divisible by 2 then it is divisible by 8.

• Both students are correct because the statements are just the converse of each other. (If p, then q ....If q then p)

• Neither students are correct because the statements are just the converse of each other. (If p, then q ....If q then p)

• Sarah M must be correct because 24 is a multiple of 8 and 2.

• Sarah R must be correct because 4 is a multiple of 2 and 8.

Tags

CCSS.6.NS.B.4

Griffin and Lindsey have been battling it out all day over the words equal and equivalent. Griffin takes his knowledge of sets and states the words have different meanings. Lindsey takes her knowledge of mathematical linguistics and states that Griffin is correct that they have different meanings. Except that he has his meanings flipped.

• Griffin gives an example where two sets are equal when everything in each set is identical; my 4 baseball buddies in set A and my same 4 baseball buddies in set B. If he had his 4 baseball buddies in set A and in set B he had his 4 math friends, these sets would be equivalent.

• Lindsey gives an example where two sets are equal in quantity; her 4 volleyball friends in set A and her 4 dance friends in set B. Equivalent, she states, must have identical sets like her 4 volleyball friends in set A and her same 4 volleyball friends in set B.

• Both students need some sleep because the words mean the same thing. The words equal and equivalent can be used interchangeably, just like math and mathematical.

• Both students need some sleep because they learned this concept back in kindergarten that equal and equivalent mean the same thing when they were playing with the blocks.

Tags

CCSS.6.EE.A.3

CCSS.6.EE.A.4

Mrs. J gives her students compliments every day. However, when it comes to complements in set theory she uses the word prime by mistake. The prime mark is really an apostrophe.

• The complement of a set A is everything in set A. It is denoted by using the apostrophe mark above the letter A to represent the complement of a set A. The word prime is only used in regards to prime numbers like the number 13.

• The words compliment and complement do not mean the same thing in set theory. The complement of a set A is the set of all elements in U, the universal set, which are not in set A. We write this with the prime mark above the letter A. The word compliment in set theory means the opposite of complement.

• The complement of a set A is the set of all elements in U, the universal set, which are not in set A. We write this with the prime mark above the letter A.

• The words compliment and complement mean the same thing in set theory. The complement of a set A is the set of all elements in U, the universal set, which are not in set A. We write this with the prime mark above the letter A.

Marissa has mastered the tough math topic of binary operations on sets. She has discovered that there is a simple way to determine if a set satisfies the commutative property.

• Marissa just looks at the Cayley table to see if it is symmetric across the diagonal.

• Marissa is mistaken, there is no easy way to look at a Cayley table to determine if the commutative property is satisfied on a set. This trick only works to check for the identity element.

• Marissa has a slight error in her reasoning. In order to check if a set satisfies the commutative property she needs to check if the set has a binary operation on the set that satisfies 3 criteria. She can do this by checking if the set is: defined, uniquely defined, closed.

• Marissa discovered that she can just look in the Cayley table to see if any square is left blank.