Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Session 6:
Homework

 Problem H1 Prime numbers have exactly two factors. Now find some numbers that have exactly three factors. What do these numbers have in common? That is, how would you categorize these numbers?

 Look for numbers with three factors, not three prime factors. The number itself and 1 are always factors, so there must be exactly one other factor. When we factor a number, we typically get two distinct factors. How could we get only one new factor?   Close Tip Look for numbers with three factors, not three prime factors. The number itself and 1 are always factors, so there must be exactly one other factor. When we factor a number, we typically get two distinct factors. How could we get only one new factor?

Problem H2

There is a way to find the number of factors of a positive integer without writing out all the factors, and it requires finding the prime factorization first. This problem will help you discover that rule.

Go through the table, and list all the factors for each number. Then in the table enter the total number of factors (including the number itself and 1). Look for patterns, and try to write a general rule for the number of factors for any integer.

Integer

Prime Factori-
zation

Number of Factors

 2 21 2 4 22 8 23 16 24 2n 2n 3 31 2 9 32 27 33 81 34 3m 3m 6 21 • 31 4 12 22 • 31 18 21 • 32 36 22 • 32 2n • 3m 2n • 3m

Integer

Prime Factori-
zation

Number of
Factors

 2 21 2 4 22 3 8 23 4 16 24 5 2n 2n (n + 1) 3 31 2 9 32 3 27 33 4 81 34 5 3m 3m m + 1 6 21 • 31 4 12 22 • 31 6 (i.e., 3 • 2) 18 21 • 32 6 (i.e., 2 • 3) 36 22 • 32 9 (i.e., 3 • 3) 2n • 3m 2n • 3m (n + 1) • (m + 1)

 Problem H3 A number is called a perfect number if the sum of all of its factors is equal to twice the value of the number. What are the two smallest perfect numbers?

 Problem H4 An abundant number is one in which the sum of its factors is greater than twice the number. A deficient number is one in which the sum of its factors is less than twice the number. Which numbers less than 25 are abundant and which are deficient?

 Problem H5 You have seen that every prime number greater than 3 is one less or one more than a multiple of 6. It is also true that every prime number greater than 2 is one more or one less than a multiple of 4. How would you prove this fact?

 Session 6: Index | Notes | Solutions | Video