Question

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?

Answer 1

Answer:

129

Step-by-step explanation:

Let a and b be two numbers.

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 $GCD(a,b)=3$.

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

Now, we will use property $GCD(x,y)*LCM(x,y)=xy$.

Upon substituting our given values, we will get:

$3*12a=ab$

$36a=ab$

Switch sides:

$ab=36a$

$\frac{ab}{a}=\frac{36a}{a}$

$b=36$

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:

$36+93=129$

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