3.4 The greatest common factor

12 and 18

1. $\begin{array}{c}\text{12}=2\cdot 6=2\cdot 2\cdot 3=2^2\cdot 3\\ \text{18}=2\cdot 9=2\cdot 3\cdot 3=2\cdot 3^2\end{array}$

2. The common bases are 2 and 3.
3. The smallest exponents appearing on 2 and 3 in the prime factorizations are, respectively, 1 and 1 ( $2^1$ and $3^1$ ), or 2 and 3.
4. The GCF is the product of these numbers.

   $2\cdot 3=6$

The GCF of 30 and 42 is 6 because 6 is the largest number that divides both 30 and 42 without a remainder.

18, 60, and 72

1. $\begin{array}{c}\text{18}=2\cdot 9=2\cdot 3\cdot 3=2\cdot 3^2\hfill \\ \text{60}=2\cdot \text{30}=2\cdot 2\cdot \mathrm{15}=2\cdot 2\cdot 3\cdot 5=2^2\cdot 3\cdot 5\hfill \\ \text{72}=2\cdot \text{36}=2\cdot 2\cdot \text{18}=2\cdot 2\cdot 2\cdot 9=2\cdot 2\cdot 2\cdot 3\cdot 3=2^3\cdot 3^2\hfill \end{array}$

2. The common bases are 2 and 3.

3. The smallest exponents appearing on 2 and 3 in the prime factorizations are, respectively, 1 and 1:

   $2^1$ from 18

   $3^1$ from 60

4. The GCF is the product of these numbers.

   GCF is $2\cdot 3=6$

Thus, 6 is the largest number that divides 18, 60, and 72 without a remainder.

700, 1,880, and 6,160 $$

1. $\begin{array}{ccccccc}\mathrm{700}& =& 2\cdot \mathrm{350}& =& 2\cdot 2\cdot \mathrm{175}& =& 2\cdot 2\cdot 5\cdot \mathrm{35}\hfill \\ & & & & & =& 2\cdot 2\cdot 5\cdot 5\cdot 7\hfill \\ & & & & & =& 2^2\cdot 5^2\cdot 7\hfill \\ \mathrm{1,880}& =& 2\cdot \mathrm{940}& =& 2\cdot 2\cdot \mathrm{470}& =& 2\cdot 2\cdot 2\cdot \mathrm{235}\hfill \\ & & & & & =& 2\cdot 2\cdot 2\cdot 5\cdot \mathrm{47}\hfill \\ & & & & & =& 2^3\cdot 5\cdot \mathrm{47}\hfill \\ \mathrm{6,160}& =& 2\cdot \mathrm{3,080}& =& 2\cdot 2\cdot \mathrm{1,540}& =& 2\cdot 2\cdot 2\cdot \mathrm{770}\hfill \\ & & & & & =& 2\cdot 2\cdot 2\cdot 2\cdot \mathrm{385}\hfill \\ & & & & & =& 2\cdot 2\cdot 2\cdot 2\cdot 5\cdot \mathrm{77}\hfill \\ & & & & & =& 2\cdot 2\cdot 2\cdot 2\cdot 5\cdot 7\cdot \mathrm{11}\hfill \\ & & & & & =& 2^4\cdot 5\cdot 7\cdot \mathrm{11}\hfill \end{array}$

2. The common bases are 2 and 5

3. The smallest exponents appearing on 2 and 5 in the prime factorizations are, respectively, 2 and 1.

   $2^2$ from 700.

   $5^1$ from either 1,880 or 6,160.

4. The GCF is the product of these numbers.

   GCF is $2^2\cdot 5=4\cdot 5=\text{20}$

Thus, 20 is the largest number that divides 700, 1,880, and 6,160 without a remainder.