The greatest common divisor of two positive integers less than $100$ is equal to $3$. Their least common multiple is twelve times one of the integers. What is the largest possible sum of the two integers?

We have been given that the greatest common divisor of two positive integers less than 100 is equal to 3. We can represent this information as [tex]GCD(a,b)=3[/tex].

Their least common multiple is twelve times one of the integers. We can represent this information as [tex]LCM(a,b)=12a[/tex].

Now, we will use property [tex]GCD(x,y)*LCM(x,y)=xy[/tex].

Upon substituting our given values, we will get:

[tex]3*12a=ab[/tex]

[tex]36a=ab[/tex]

Switch sides:

[tex]ab=36a[/tex]

[tex]\frac{ab}{a}=\frac{36a}{a}[/tex]

[tex]b=36[/tex]

Now, we need to find a number less than 100, which is co-prime with 12 after dividing by 3.

The greatest multiple of 3 less than 100 is 99, but it is not co-prime with 12 after dividing by 3.

Similarly 96 is also not co-prime with 12 after dividing by 3.

We know that greatest multiple of 3 (less than 100), which is co-prime with 12, is 93.

Let us add 36 and 93 to find the largest possible sum of the required two integers as:

[tex]36+93=129[/tex]

Therefore, the required largest possible sum of the two integers is 129.