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ACTIVITY 2
The Primes of Your Life
Lesson 2-1 Prime Factorization
Learning Targets:
Determine whether a given whole number is a prime number or a
composite number.
Express a composite number as a product of prime numbers.
My Notes
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SUGGESTED LEARNING STRATEGIES: Create Representations,
Note Taking, Think-Pair-Share, Visualization, Sharing and Responding
The prime factorization of a number shows the number as a product of
factors that are all prime numbers. One way to find the prime
factorization of a number is to use a visual model called a factor tree.
A prime factor is a factor that is
prime.
Find the prime factorization of 12.
Use divisibility rules to find two factors of
the number.
Try 3 and 4. Use branches to show the
factors.
Step 2:
A prime number is a natural
number greater than 1 that has
exactly two factors, itself and 1.
For example, 2 is a prime number
(2 = 2 × 1), as is 13 (13 = 13 × 1).
A factor is one of the numbers you
multiply to get a product.
Example A
Step 1:
MATH TERMS
If both numbers are prime, stop. If not,
continue factoring until all factors are prime
numbers.
12
3×4
A composite number is a natural
number that has more than two
factors. For example, 15 is a
composite number (15 = 1 × 3 × 5).
3×2×2
3 is prime. Bring it down to the next branch.
Continue by factoring 4.
Step 3:
Check again to be sure all factors are now prime numbers.
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3=3×1✓
2=2×1✓
The only factors of 3 and 2 are 1 and the numbers themselves,
so 3 and 2 are prime.
Solution: The prime factorization of 12 is 3 × 2 × 2.
Try These A
a. Reason quantitatively. Will the prime factorization of 12 be
different if you start with the factors 2 and 6? Explain.
Find the prime factorization of each number.
b. 21
c. 16
d. 18
Activity 2 • Prime Factorization and Exponents
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Prime Factorization
ACTIVITY 2
continued
My Notes
Here is another method for finding prime factors.
Example B
Find the prime factorization of 60.
Step 1: Write the number as the
dividend inside a division
symbol. Write one of the
prime factors as the divisor
on the outside.
Prime factorizations are usually
written with the factors ordered
from least to greatest:
2 × 3 × 3 × 7, not 3 × 7 × 3 × 2
Step 2:
Divide as you would if you were using long division.
Step 3:
Repeat the steps, this time using the quotient on top of the
division symbol as the new dividend.
Step 4:
Stop when the quotient is a prime number.
Step 5:
Use the divisors and the final quotient to write the prime
factorization.
Solution: 60 = 2 × 2 × 3 × 5
Try These B
Find each prime factorization using the long division method.
a. 32
b. 45
c. 56
Even if a number is not divisible by any small natural numbers such as 2,
3, or 5, it may still be a composite number. To find its prime factorization,
you may have to use larger numbers to guess and check with.
Example C
At After School Sports Club, 143 students are divided into teams, with
the same number on each team. How many teams are there and how
many students are on each team?
To solve, use divisibility rules to see if 143 is divisible by any prime numbers.
Start with 2 and work upwards.
2? No. 143 is not even.
3? No. 1 + 4 + 3 = 8, which is not divisible by 3.
5? No. The ones digit of 143 is not 0 or 5.
7? No. When you divide 143 by 7, there is a remainder.
11? Yes. 143 ÷ 11 = 13.
Solution: Since 143 ÷ 11 = 13, 143 = 11 × 13. That means there could
by 11 teams with 13 on each team, or 13 teams with 11 on
each team.
28 Unit 1 • Number Concepts
© 2014 College Board. All rights reserved.
WRITING MATH
5 ←Step 4
2 10
3 30 ←Steps 2 & 3
2
60 ←Step 1
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Prime Factorization
ACTIVITY 2
continued
My Notes
Try These C
a. Last year there were 133 students in After School Sports Club. How
many teams were there, and how many students were on each team?
b. There are 221 math books in a closet arranged in equal stacks. How
many stacks are there, and how many books are in each stack?
Check Your Understanding
1. Determine the prime factorization of each number.
a. 14
d. 38
g. 100
b. 30
e. 84
h. 77
c. 27
f. 41
i. 180
2. Why is every prime number greater than 2 an odd number?
3. Explain why numbers with a 5 in the ones place are not prime
numbers.
4. List all the prime numbers from 1 to 50.
5. Explain why you cannot find the prime factorization of 4.8.
© 2014 College Board. All rights reserved.
/(6621 35$&7,&(
6. Construct viable arguments. 7,719 = 83 × 93. Explain why 83 × 93
is not the prime factorization of 7,719.
7. A conjecture is a statement that appears to be true, but which remains
unproven. In 1976, a seventh-grade student named Arthur Hamann
made a conjecture that every even number can be expressed as the
difference between two prime numbers.
6 = 19 − 13
20 = 23 − 3
ACADEMIC VOCABULARY
A conjecture is a statement that
appears to be true but has not
been proven.
No one has ever found an even number for which Arthur Hamann’s
conjecture is not true. Test the conjecture for these even numbers:
a. 14
b. 18
c. 22
8. A famous conjecture by the eighteenth-century mathematician
Christian Goldbach also remains unproven. Goldbach’s conjecture
states that every even number greater than 2 can be expressed as the
sum of two prime numbers.
12 = 5 + 7
26 = 13 + 13
Test Goldbach’s conjecture for these even numbers:
a. 16
b. 26
c. 34
Activity 2 • Prime Factorization and Exponents
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