Basic Mathematics for College Students

84 Chapter 1 Whole Numbers

Caution! Remember that there is a difference between the factorsand the

prime factorsof a number. For example,

The factors of 15 are:

The prime factors of 15 are: 3 5

1, 3, 5, 15

4 Find prime factorizations using a division ladder.

We can also find the prime factorization of a whole number using an inverted

division process called a division ladder.It is called that because of the vertical

“steps” that it produces.

Success Tip The divisibility rules found in Section 1.5 are helpful when using

the division ladder method. You may want to review them at this time.

EXAMPLE (^6) Use a division ladder to find the prime factorization of 280.
StrategyWe will perform repeated divisions by prime numbers until the final
quotient is itself a prime number.
WHYIf a prime number is a factor of 280, it will divide 280 exactly.
Solution
It is helpful to begin with the smallest prime,2, as the first trial divisor. Then, if
necessary, try the primes in that order.
Step 1The prime number 2 divides 280 exactly.
The result is 140, which is not prime. Continue the division
process.
Step 2Since 140 is even, divide by 2 again.
The result is 70, which is not prime. Continue the division
process.
Step 3Since 70 is even, divide by 2 a third time. The result
is 35, which is not prime.
Continue the division process.
Step 4Since neither the prime number 2 nor the next greatest
Prime
prime number 3 divide 35 exactly, we try 5. The result is 7, which
is prime. We are done.
The prime factorization of 280 appears in the left column of the
division ladder: Check this result using
multiplication.

2  2  2  5 7. 

2  280

2  140

(^2)  70
5  35
7
(^2)  280
2  140
(^2)  70
35
(^2)  280
2  140
70
(^2)  280
140
3, 5, 7, 11, 13, p
Self Check 6
Use a division ladder to find the
prime factorization of 108.
Now TryProblems 63 and 73
Caution! In Example 6, it would be incorrect to begin the division process
with
because 4 is not a prime number.

4 280

70