52 Part 1: The World of Numbers
Using Exponents
Some prime factorizations are easy to write. The prime factorization of 42 is 2 v 3 v 7,
which is short and clear. However, the prime factorization of 128 is another story. 128 =
2 v 2 v 2 v 2 v 2 v 2 v 2. It uses easy numbers, but it uses so many of them that it can get
confusing. Is that six 2s or seven? It’s easy to make a mistake.
That’s why you may want to use exponents to condense the prime factorization and make it
easier to understand. The prime factorization of 40 = 2 v 2 v 2 v 5, but it could be written
as 40 = 2^3 v 5. The exponent of 3 says to use 2 as a factor 3 times. With that notation, you can
write the prime factorization of 128 as 128 = 2^7 and know for certain that it was seven 2s.CHECK POINT
Use exponents to write the prime factorization of each number in a compact form.- 20 0
- 168
- 672
14. 2,205
15. 22,000
Finding the prime factorization of a number can be an interesting undertaking. It can be difficult,
but there’s a satisfaction in getting a large number broken down to a product of primes. The truth,
however, is that finding a prime factorization is not usually your final goal, but a tool to help you
find two other very useful numbers: the greatest common factor and the least common multiple.Greatest Common Factor
The greatest common factor, or GCF, of two numbers is the largest number that is a factor of
both. The greatest common factor of 21 and 24 is 3. The factors of 21, other than 1 and 21, are 3
and 7. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The largest number on both lists is 3.DEFINITION
The greatest common factor (GCF) of two numbers is the largest number that is a
factor of both.To find the greatest common factor of 70 and 105, first find the factors of 70. The factors of 70
are 1, 2, 5, 7, 10, 14, 35 and 70. Then find the factors of 105. Those are 1, 3, 5, 7, 15, 21, 35, and 105.
The largest number that shows up on both factor lists is 35, so 35 is the GCF of 70 and 105.