SAT Math : How to find the least common multiple
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Example Question #71 : Integers
What is the least common multiple of 
We need to ensure that all the numbers share a common factor of 
Example Question #11 : How To Find The Least Common Multiple
What is the least common multiple of 
We need to ensure that all the numbers share a common factor of 
Example Question #11 : Least Common Multiple
What is the least common multiple of the first six positive integers?
Let's divide the even numbers first. We will divide them by 
Next, we have two 
Example Question #11 : Least Common Multiple
Which can be a group of remainders when four consecutive integers are divided by 
If you divide a number by 
Example Question #11 : Least Common Multiple
If a, b, and c are positive integers such that 4a = 6b = 11c, then what is the smallest possible value of a + b + c?
33
132
67
11
121
67
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.
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