Number Theory Problem Solving

Math 123

Number theory problem solving

1. Dominic's Automobile Agency has 80 new cars lined up in their lot. Every third car has a vinyl roof. Every fourth car has a stick shift, and every eighth car has a security system. Audrey wants to buy a car with all three of these options. How many cars at Dominic's will meet her needs?

2. The Stewarts are buying cups and plates for their huge annual family picnic. Cups come in packages of 54, whereas plates come in packages of 42. How many packages of each must the Stewarts buy to have the same number of cups and plates?

3. Sarah has between 50 and 100 pennies in her collection. When she divides them into groups of 2, of 3, or of 7, there is always 1 penny left. How many pennies does Sarah have in her collection? Write to help explain your best thinking using words, numbers, or pictures.

4. At the Bubble Gum Factory, lengths of gumare stretched to larger lengths by putting them through stretchingmachines. There are 100 stretching machines, numbered 1 through 100.Machine 1 does nothing to a piece of gum; machine 2 stretches piecesof gum to twice their original length; machine 3 triples the lengthof gum, and so forth. So, machine 23, for example, will stretch apiece of gum to 23 times its original length.

An order has just come in for a piece of bubble gum 26 inches inlength. The factory has pieces of gum that are only 1 inch inlength, and machine number 26 is broken. Is there any way to createa piece of bubble gum 26 inches in length by using other machines?

Some of the machines in the factory are unnecessary, becausecombinations of others machines could be used instead. Figure outwhich machines are actually unnecessary.

What machines would be necessary to get the followinglengths: 15? 28? 36? 65? 84?

Which lengths between 1 and 100 would come out if the bubblegum went through five machines and all five machines were necessaryones?

Which lengths between 1 and 100 require the greatest number ofnecessary machines? How did you figure out your answer?

5. Andrew and Bert met on the street and had the followingconversation:

Andrew: How old are your three children?

Bert: The product of theirages is 36.

Andrew: That's not enough information for me to knowtheir ages.

Bert: The sum of their ages is your house number.

Andrew: That's still not quite enough information.

Bert: The oldestchild plays the piano.

Andrew: Now I know!

6. 1000 students areeach assigned 1 of 1000 lockers. The first day of school, all thelockers are closed and the students line up outside the school innumerical order. student 1 opens all the lockers since all of themare multiples of 1. Student #2 closes all the lockers that aremultiples of 2. Student #3 goes to lockers 3,6, 9 and so on andchanges the position of each (opening the ones that are closed, andclosing the ones that are open). Each subsequent student does thesame to the multiples of his/her number. At the end, how manylockers are open and how many are closed?