### Instruction: Simplify $\frac{84}{144}.$.### Response: I want to simplify this fraction by finding a common factor of both the numerator and the denominator.
A common factor is a number that divides both 84 and 144 evenly, without leaving any remainder.
I can start by checking some small numbers, like 2, 3, 4, and so on, and see if they are common factors.
I notice that both 84 and 144 are even, which means they are divisible by 2.
So I can divide both the numerator and the denominator by 2 to get a smaller fraction.
$\frac{84}{144}=\frac{84\div 2}{144\div 2}=\frac{42}{72}.$
I'm not done yet, because I can simplify this fraction further.
I see that both 42 and 72 are still even, so I can divide them by 2 again.
$\frac{42}{72}=\frac{42\div 2}{72\div 2}=\frac{21}{36}.$
Now I check if there are any other common factors of 21 and 36.
I know that 21 is divisible by 3, because 3 times 7 is 21.
But is 36 divisible by 3?
I can use a trick to check if a number is divisible by 3.
I add up the digits of the number, and if the sum is divisible by 3, then the number is divisible by 3 as well.
The digits of 36 are 3 and 6, and their sum is 9.
9 is divisible by 3, so 36 is divisible by 3 as well.
That means I can divide both 21 and 36 by 3 to simplify the fraction further.
$\frac{21}{36}=\frac{21\div 3}{36\div 3}=\frac{7}{12}.$
Now I have a simplified fraction, and I can't simplify it any more, because 7 and 12 have no common factors other than 1.