Section 1
Prime Numbers and Divisibility
List the factors of each number.$$4$$
List the factors of each number.$$6$$
List the factors of each number.$$10$$
List the factors of each number.$$12$$
List the factors of each number.$$15$$
List the factors of each number.$$21$$
List the factors of each number.$$24$$
List the factors of each number.$$32$$
List the factors of each number.$$64$$
List the factors of each number.$$66$$
List the factors of each number.$$11$$
List the factors of each number.$$37$$
List the factors of each number.$$135$$
List the factors of each number.$$236$$
List the factors of each number.$$256$$
List the factors of each number.$$512$$
Use the list of numbers$$0,1,15,19,23,31,49,55,59,87,91,97,103,105$$Which of the given numbers are prime?
Use the list of numbers$$0,1,15,19,23,31,49,55,59,87,91,97,103,105$$Which of the given numbers are composite?
List all the prime numbers between 30 and 50 .
List all the prime numbers between 55 and 75 .
Use the list of numbers$$45,72,158,260,378,569,570,585,3,541,4,530,8,300$$Which of the given numbers are divisible by $2 ?$
Use the list of numbers$$45,72,158,260,378,569,570,585,3,541,4,530,8,300$$Which of the given numbers are divisible by $3 ?$
Use the list of numbers$$45,72,158,260,378,569,570,585,3,541,4,530,8,300$$Which of the given numbers are divisible by $6 ?$
Use the list of numbers$$45,72,158,260,378,569,570,585,3,541,4,530,8,300$$Which of the given numbers are divisible by $9 ?$
Use the list of numbers$$45,72,158,260,378,569,570,585,3,541,4,530,8,300$$Which of the given numbers are divisible by $4 ?$
Use the list of numbers$$45,72,158,260,378,569,570,585,3,541,4,530,8,300$$Which of the given numbers are divisible by $10 ?$
A school auditorium is to have 350 seats. The principal wants to arrange them in rows with the same number of seats in each row. Use divisibility tests to determine if it is possible to have rows of 10 seats each. Are 15 rows of seats possible?
Dr. Mento has a class of 80 students. For a group project, she wants to divide the students into groups of $6,8,$ or $10 .$ Is this possible? Explain your answer.
Use the sieve of Eratosthenes to determine all the prime numbers less than 100 .(TABLE CAN NOT COPY)
Why is the statement not a valid divisibility test for $8 ?$"A number is divisible by 8 if it is divisible by 2 and 4 ." Support your answer with an example. Give a valid divisibility test for $8 .$
Prime numbers that differ by 2 are called twin primes. Examples are 3 and 5 , 5 and $7,$ and so on. Find one pair of twin primes between 85 and 105
The following questions refer to twin primes (see exercise 31 ).(a) Search for, and make a list of, several pairs of twin primes in which the primes are greater than 3(b) What do you notice about each number that lies between a pair of twin primes?(c) Write an explanation for your observation in part (b).
Obtain (or imagine that you have) a quantity of square tiles. Six tiles can be arranged in the shape of a rectangle in two different ways:(FIGURE CAN NOT COPY)(a) Record the dimensions of the rectangles shown.(b) If you use 7 tiles, how many different rectangles can you form?(c) If you use 10 tiles, how many different rectangles can you form?(d) What kind of number (of tiles) permits only one arrangement into a rectangle? More than one arrangement?
The number 10 has 4 factors: $1,2,5,$ and $10 .$ We can say that 10 has an even number of factors. Investigate several numbers to determine which numbers have an even number of factors and which numbers have an odd number of factors.
Suppose that a school has $1,000$ lockers and that they are all closed. A person passes through, opening every other locker, beginning with locker $2 .$ Then another person passes through, changing cvery third locker (closing it if it is open, opening it if it is closed), starting with locker $3 .$ Yet another person passes through, changing every fourth locker, beginning with locker $4 .$ This process continues until $1,000$ people pass through. (a) At the end of this process, which locker numbers are closed?(b) Write an explanation for your answer to part (a). (Hint: It may help to attempt exercise 34 first.)(IMAGE CAN NOT COPY)