SAT Math : How to find the least common multiple

Study concepts, example questions & explanations for SAT Math

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Example Questions

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Example Question #1 : How To Find The Least Common Multiple

How many positive integers less than ten thousand are multiples of both eight and eighteen?

 

Possible Answers:

138

555

72

139

70

Correct answer:

138

Explanation:

In order to find all of the numbers that are multiples of both 8 and 18, we need to find the least common mutliple (LCM) of 8 and 18. The easiest way to do this would be to list out the multiples of 8 and 18 and determine the smallest one that is common to both. 

First, let's list the first several multiples of eight:

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88 . . .

Next, we list the first several multiples of eighteen:

18, 36, 54, 72, 90, 108, 126, 144 . . .

By comparing the multiples of eight and eighteen, we can see that the smallest one that they share is 72. Thus, the LCM of 8 and 18 is 72.

Because the LCM is 72, this means that every multiple of 72 is also a multiple of both 8 and 18. So, in order to find all of the multiples less than ten thousand that are both multiples of 8 and 18, we simply need to find how many multiples of 72 are less than 10000, and to do this, all we have to do is to divide 10000 by 72.

When we divide 10000 by 72, we get 138 with a remainder of 64; therefore, 72 will go into ten thousand 138 times before it exceeds ten thousand. In other words, there are 138 numbers less than 10000 that are multiples of 72 and, by extension, also multiples of both 8 and 18. 

The answer is 138.

Example Question #1 : How To Find The Least Common Multiple

What is the least common multiple of ?

Possible Answers:

Correct answer:

Explanation:

 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 .

Example Question #3 : How To Find The Least Common Multiple

What is the least common multiple of ?

Possible Answers:

Correct answer:

Explanation:

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. 

Example Question #2 : How To Find The Least Common Multiple

What is the least common multiple of ?

Possible Answers:

Correct answer:

Explanation:

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 

Example Question #5 : How To Find The Least Common Multiple

What is the least common multiple of ?

Possible Answers:

Correct answer:

Explanation:

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 

Example Question #6 : How To Find The Least Common Multiple

What is the least common multiple of ?

Possible Answers:

Correct answer:

Explanation:

Both numbers are divisible by , so the remaining numbers are . 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 

Example Question #5 : Least Common Multiple

What is the least common multiple of ?

Possible Answers:

Correct answer:

Explanation:

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 

Example Question #2 : How To Find The Least Common Multiple

What is the least common multiple of ?

Possible Answers:

Correct answer:

Explanation:

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 .

Example Question #7 : Least Common Multiple

What is the least common multiple of ?

Possible Answers:

Correct answer:

Explanation:

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 

Example Question #3 : How To Find The Least Common Multiple

What is the least common multiple of ?

Possible Answers:

Correct answer:

Explanation:

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 

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