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. Mrs. Taylor has a new way of giving homework problems. If the date is divisible by 2, she gives 2 problems. If the date is divisible by 3, she gives 3 problems. If the date is divisible by 5, she gives 5 problems. Otherwise, she doesn't give any problems on that day. On the 6th of the month, she gave 5 problems, since 6 is divisible by both 2 and 3. How many problems did she assign on the 10th? On which dates would there be no problems assigned?
3. 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?
4. When a man went to visit a friend, he realized he had forgotten the apartment number. He does remember the apartment is on the first floor where the apartments are numbered 101 through 150. He also knows the apartment number is an even number and a multiple of 3 and 11. What is the correct apartment number?
5. 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.
6. The number of students on the field trip is greater than 30 but less than 50. When seated 2 in a seat on the bus, no student has to sit alone. When placed in groups of 5 for a tour, all groups are the same size. How many students are on the field trip? Write to help explain your best thinking using words, numbers, or pictures.