Unit I: Prime Time
ACE Problems 3.2
Name: __________________________
Date: ___________________________
Applications
For Exercises 5, 6, 8 and 11, write the prime factorization of each number in expanded form.
5. 36
8. 165
6. 180
11. 216
14. Use exponents to rewrite the prime factorizations in Exercises 5 and 11.
15. Multiple Choice What is the prime factorization of 240?
A. 10 x 24
B. 2 x 3 x 5
C. 23 x 3 x 5
D. 24 x 3 x 5
18. Multiple Choice Choose the number that is the product of exactly three different prime
numbers.
F. 15
G. 20
H. 30
J. 57
Unit I: Prime Time
Connections
31. Rosa claims the longest string of prime factors for 30 is 2 x 3 x 5. Tyee claims there is a longer
string, 1 x 2 x 1 x 3 x 1 x 5. Who is correct? Why?
36. Suppose 10 and 6 are common factors of two numbers. What other factors must the numbers
have in common? Explain.
Unit I: Prime Time
Extra Effort
For exercises 7,9,10,12, and 13, write the prime factorization of each number in expanded
form
7. 525
9. 293
10. 760
12. 231
13. 312
16. Jill and Jamahl are comparing their favorite numbers. Jill’s favorite number has a prime
factorization with six numbers. Jamahl’s number has a prime factorization with only three numbers.
Jill says that this means her number is greater than Jamahl’s. Jamahl says that is not necessarily
true. Who is correct? Explain.
17. Find all of the numbers less than 100 that have at least one 2 and at least one 5 in their prime
factorization. What do you notice about these numbers?
19. Find all of the numbers less than 100 that are the product of exactly three different prime
numbers.
Unit I: Prime Time
20. Mr. and Mrs. Fisk have 8 children. Each of those children has 8 children.
a. How many grandchildren do Mr. and Mrs. Fisk have?
b. Each grandchild has 8 children. How many great-grandchildren do Mr. and Mrs. Fisk have?
c. Write an expression with exponents to represent the number of great-grandchildren Mr.
and Mrs. Fisk have.
32. The number 1 is not prime. Why do you think mathematicians decided not to call 1 a prime
number?
33. How many factors do each of the following numbers have?
a. 100
b. 101
c. 102
d. 103
e. What patterns do you notice in the factors of these numbers have?
For Exercises 34-35, describe the numbers that have both of the given numbers as factors.
34. 2 and 3
35. 3 and 5
Extensions
51. For each part below, use your birth year or the birth year of one of your family members.
a. Find the prime factorization of the birth year.
Unit I: Prime Time
b. Describe the number to a friend with as much information as you can about it. Here are
some ideas to include: Is the number square, prime, even, odd? How many factors does it
have? Is it a multiple of some other number?
52. Most years contain 365 days, but certain years, called leap years, contain 366 days. Leap years
occur in years divisible by 4, with some exceptions. Years divisible by 100 are not leap years—unless
they are divisible by 400. So 1896 was a leap year, but 1900 wasn’t. The years 1996, 2000, 2004,
2008, and 2012 were all leap years.
a. A week has 7 days. How many weeks are in each type of year?
b. January 1, 2012, fell on a Sunday. On what dates did the next three Sundays of 2012
occur?
c. What day of the week was January 30, 2012?
d. The year 2012 was a leap year. It had 366 days. What day of the week was January 1,
2013?
e. Find the pattern, over several years, for the days on which your birthday falls?
53. The Greek mathematician Euclid first stated the Fundamental Theorem of Arithmetic. He
wrote:
“If a number is the least that is measured by prime numbers, it will not
be measured by any prime except those originally measuring it.”
What do you suppose Euclid meant?