Both numbers are divisible by  because the sum of the digits are divisible by . We get  as the remaining numbers. We can divide by  to get . We have two prime numbers. Now, we multiply the factors and the remaining numbers to get  or .

Both  are even so we can divide both numbers by  to get . We have a prime number and a composite number respectively. They share a factor of . To determine the least common multiple, we multiply the factor with the numbers remaining. Our answer is just  or . 

We need to ensure that all the numbers share a common factor of .  are even so let's divide those by . We get  leftover along with the  that doesn't divide evenly with . Now that all these numbers share a common factor of , we multiply them all out including the  we divided out. We get  or . 

We need to ensure that all the numbers share a common factor of .  are divisible by . We get  leftover along with the  that doesn't divide evenly with . Now that all these numbers share a common factor of , we multiply them all out including the  we divided out. We get  or . 

We need to ensure that all the numbers share a common factor of .  are divisible by . We get  leftover along with the  that doesn't divide evenly with . Next,  are divisible by . We also get  leftover. Then, we can divide the s out to get . Now that all these numbers share a common factor of , we multiply them all out including the  we divided out. We get  or . 

Let's divide the even numbers first. We will divide them by .

Next, we have two s, so let's divide them by  to get . So far we have factors of  remaining from the original six integers with factors of  been used. Now that they have a common factor of , we multiply everything out. We get  or . 

If you divide a number by , you cannot have a remainder of   You can either have  in that order.

We are told that a, b, and c are integers, and that 4a = 6b = 11c. Because a, b, and c are positive integers, this means that 4a represents all of the multiples of 4, 6b represents the multiples of 6, and 11c represents the multiples of 11. Essentially, we will need to find the least common multiples (LCM) of 4, 6, and 11, so that 4a, 6b, and 11c are all equal to one another.

First, let's find the LCM of 4 and 6. We can list the multiples of each, and determine the smallest multiple they have in common. The multiples of 4 and 6 are as follows:

4: 4, 8, 12, 16, 20, ...

6: 6, 12, 18, 24, 30, ...

The smallest multiple that 4 and 6 have in common is 12. Thus, the LCM of 4 and 6 is 12.

We must now find the LCM of 12 and 11, because we know that any multiple of 12 will also be a multiple of 4 and 6.

Let's list the first several multiples of 12 and 11:

12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, ...

11: 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, ...

The LCM of 12 and 11 is 132.

Thus, the LCM of 4, 6, and 12 is 132.

Now, we need to find the values of a, b, and c, such that 4a = 6b = 12c = 132.

4a = 132

Divide each side by 4.

a = 33

Next, let 6b = 132.

6b = 132

Divide both sides by 6.

b = 22

Finally, let 11c = 132.

11c = 132

Divide both sides by 11.

c = 12.

Thus, a = 33, b = 22, and c = 12.

We are asked to find the value of a + b + c.

33 + 22 + 12 = 67.

The answer is 67.

Since v is divisible by 2, 3 and 15, v must be a multiple of 30. Any number that is divisible by both 2 and 15 must be divisible by their product, 30, since this is the least common multiple.

Out of all the answer choices, v + 30 is the only one that equals a multiple of 30.

We are given information that m/4 is 10 greater than m/3. We set up an equation where m/4 = m/3 + 10.

We must then give the m variables a common denominator in order to solve for m. Since 3 * 4 = 12, we can use 12 as our denominator for both m variables.

m/4 = m/3 + 10 (Multiply m/4 by 3 in the numerator and denominator.)

3m/12 = m/3 + 10 (Multiply m/3 by 4 in the numerator and denominator.)

3m/12 = 4m/12 + 10 (Subtract 4m/12 on both sides.)

-m/12 = 10 (Multiply both sides by -12.)

m = -120

-120/4 = -30 and -120/3 = -40. -30 is 10 greater than -40.