Least common multiple is the smallest number that is divisible by two or more factors. Since are prime numbers and can't be broken down to smaller factors, we just multiply them to get as our answer.

We are inclined to multiply the numbers out however, if we divide both numbers by , we get remaining. These numbers are unit and prime numbers respectively and only share a factor of . To determine the least common multiple, we multiply the factor with the numbers remaining. Our answer is just 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 .

are different kind of numbers. We have a composite number and a prime number, respectively. They share a factor of . Therefore, we just multiply both numbers to get an answer of .

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 are told that a, b, and c are integers, and that 4_a_ = 6_b_ = 11_c_. Because a, b, and c are positive integers, this means that 4_a_ represents all of the multiples of 4, 6_b_ represents the multiples of 6, and 11_c_ represents the multiples of 11. Essentially, we will need to find the least common multiples (LCM) of 4, 6, and 11, so that 4_a_, 6_b_, and 11_c_ 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 4_a_ = 6_b_ = 12_c_ = 132.

4_a_ = 132

Divide each side by 4.

a = 33

Next, let 6_b_ = 132.

6_b_ = 132

Divide both sides by 6.

b = 22

Finally, let 11_c_ = 132.

11_c_ = 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.

If we divide for both numbers, we get . We do it the second time and we get . Now we have a unit and a prime number. So we just multiply the factors and the remaining numbers to get or .

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 .

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