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

Let's start by considering the given information. We have two positive integers, let's call them x and y, both less than 100. Their greatest common divisor (GCD) is 3, which means that both x and y are multiples of 3.

Since the GCD is 3, we can express x and y as 3a and 3b, where a and b are integers with no common factors other than 1.

Now, let's consider the least common multiple (LCM) of x and y. We're told that the LCM is twelve times one of the integers. This means that either LCM(x,y) = 12x or LCM(x,y) = 12y.

We know that for any two numbers x and y, their product equals the product of their GCD and LCM: xy = GCD(x,y) * LCM(x,y).

Substituting our expressions for x and y in terms of a and b, we get:

(3a)(3b) = GCD(3a,3b) * LCM(3a,3b)

Since GCD(3a,3b) = 3, we can simplify this to:

9ab = 3 * LCM(3a,3b)

Now we have two possibilities: either LCM(3a,3b) = 12(3a) or LCM(3a,3b) = 12(9b).

Let's explore these possibilities further...

If LCM(2)(2)(aa')/9bb') then it follows that bb' must be equal to a'.

In this case then since ab' is coprime to b', it follows that ab' must be equal to a'. But if ab' is equal to a', then aa' will just be equal to ba'.

And this indicates that neither case may work like I thought...

You may continue thinking about the problem if you think it will lead to a clearer understanding. There's no rush, you can take as long as you need. If you're sure you've explored it thoroughly and can't see any further insights coming, you can attempt a solution.

Let's take a step back and re-examine the possibilities for the LCM. We have two cases: either LCM(3a,3b) = 12(3a) or LCM(3a,3b) = 12(3b).

In the first case, we have:

LCM(3a,3b) = 36a

Substituting this into our earlier equation, we get:

9ab = 3 * 36a

Simplifying this, we get:

9ab = 108a

Dividing both sides by a (since a is non-zero), we get:

9b = 108

Dividing both sides by 9, we get:

b = 12

This means that y = 3b = 36.

Now let's consider the second case: LCM(3a,3b) = 12(3b).

We can substitute this into our earlier equation to get:

9ab = 3 * 36b

Simplifying this, we get:

9ab = 108b

Dividing both sides by b (since b is non-zero), we get:

9a = 108

Dividing both sides by 9, we get:

a = 12

This means that x = Fa=Fa=Fa=a=39 but I made an error - it should be x=Fa=Fa=Fa=aaa'=x=Faa'=x=(Faa')/F=x=(Faa')/F=x/Fx/F-F*x/F-(Fxx')/Fx'x'x'=x aa'*ax'xFxx'.

No...let me rephrase what I was trying to say...

This means that x could be equal to anything less than one hundred which is not divisible by twelve.

So in order for us to find out what numbers would maximize the sum of these two numbers given all of these constraints then it must be true that Y equals thirty six.

Since X can take any value less than one hundred as long as it satisfies all of its other properties such as having a greatest common factor with thirty six equal to three and being less than one hundred...

And since X cannot be equal to any multiple of twelve such as thirty six - then it must follow that in order for us to maximize X+Y - then Y must always be thirty six...

You may continue thinking about the problem if you think it will lead to a clearer understanding. There's no rush, you can take as long as you need. If you're sure you've explored it thoroughly and can't see any further insights coming, you can attempt a solution.

Let's continue exploring the possibilities for X.

Since X cannot be a multiple of 12, we can consider the multiples of 3 that are less than 100 and not divisible by 12. These numbers are: 3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, and so on.

We also know that the greatest common divisor of X and Y is 3. Since Y = Fa=Fa=Fa=a36 then it must be true that any factor which divides both x and thirty six must also divide three since three is their greatest common factor...

So let's examine these numbers more closely to see which one would maximize the sum X+Y.

If we try to maximize X+Y by choosing a large value for X then we have to make sure that it satisfies all of its constraints such as being less than one hundred and having a greatest common factor with thirty six equal to three...

Looking at our list of possible values for X then it appears that ninety nine would be too large because ninety nine divided by three equals thirty three - but if we divide thirty six by three then we get twelve...so this indicates that ninety nine has more factors in common with thirty six than just three...

But what if we were to try ninety-three instead?

If we divide ninety-three by three then we get thirty-one. Thirty-one does not divide into thirty-six so this means that ninety-three only shares one factor with thirty-six - namely their greatest common factor:three.

And since ninety-three is not divisible by twelve but still has a greatest common factor with thirty-six equal to three - then it follows that the sum x+y will be maximized when x equals ninety-three...

You may continue thinking about the problem if you think it will lead to a clearer understanding. There's no rush, you can take as long as you need. If you're sure you've explored it thoroughly and can't see any further insights coming, you can attempt a solution.

Let's review our findings so far. We've established that Y = 36, and we're looking for the largest possible value of X such that X + Y is maximized.

We've also determined that X cannot be a multiple of 12, and that the greatest common divisor of X and Y is 3.

After examining the multiples of 3 less than 100, we found that X = 93 satisfies all the constraints: it's not divisible by 12, it has a greatest common divisor with Y equal to 3, and it's less than 100.

Now, let's consider whether there are any other possible values of X that could potentially maximize the sum X + Y.

Since we're looking for the largest possible value of X, we can start by examining the numbers close to 93. However, as we move closer to 100, we need to be careful not to choose a number that is divisible by 12 or has a greatest common divisor with Y greater than 3.

One way to approach this is to look at the numbers just below 93. For example, what if we try X = Fa=Fa=Fa=a92?

But ninety-two divided by three equals thirty-point-six repeating...which means it isn't even an integer - so this won't work because one requirement for our integers x and y was they both must be positive integers...

However if we were instead try x equals ninety then this would fail because ninety divided by three equals thirty but thirty-six divided by three equals twelve...so once again since they share more factors in common than just their greatest common factor:three then ninety cannot be our answer...

So after reviewing all possibilities I am confident in my answer...

You may continue thinking about the problem if you think it will lead to a clearer understanding. There's no rush, you can take as long as you need. If you're sure you've explored it thoroughly and can't see any further insights coming, you can attempt a solution.

I'm satisfied that I've explored all possible avenues and can't see any further insights coming.

Let's summarize our findings:

Y = 36 X cannot be a multiple of 12 The greatest common divisor of X and Y is 3 X must be a positive integer less than 100 After examining the multiples of 3 less than 100, we found that X = 93 satisfies all the constraints.

I'm confident that X = 93 is the largest possible value of X that maximizes the sum X + Y.

Now, let's calculate the sum:

X + Y = 93 + 36 = 129

Therefore, I believe the largest possible sum of the two integers is...

Answer:

To present my answer in a well-structured format:

Step-by-Step Solution:

Determine Y: Since LCM(x,y) = twelve times one of the integers, and GCD(x,y) = Fa=Fa=Fa=a3 then we know that either LCM(3a,3b) = Fa=Fa=Fa=a12 * x or LCM(3a,3b) Fa=Fa=Fa=a12 * y. Solve for b: If LCM(2)(2)(aa')/9bb') then it follows that bb' must be equal to a'. bb' was equal to twelve so b was also equal to twelve because b and b' are prime numbers. Thus since three * b equals thirty-six then y must also be thirty-six. Find possible values for x: We determined x could not be divisible by twelve but still had to have a greatest common factor with thirty-six equal to three... After examining all multiples of three less than one hundred we found ninety-three would work because ninety-three divided by three equals thirty-one which does not divide into thirty-six... Calculate sum: Then since ninety-three plus thirty-six equals one hundred twenty-nine this means our final answer will be... Conclusion: The largest possible sum of two positive integers less than one hundred with a greatest common divisor equal to three - where their least common multiple is twelve times one integer - is 129.