Least Common Multiple - SAT Math
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How many positive integers less than ten thousand are multiples of both eight and eighteen?
How many positive integers less than ten thousand are multiples of both eight and eighteen?
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.
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.
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If a, b, and c are positive integers such that 4_a_ = 6_b_ = 11_c_, then what is the smallest possible value of a + b + c?
If a, b, and c are positive integers such that 4_a_ = 6_b_ = 11_c_, then what is the smallest possible value of a + b + c?
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.
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.
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What is the least common multiple of 
?
What is the least common multiple of
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.
Least common multiple is the smallest number that is divisible by two or more factors. Since
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What is the least common multiple of 
?
What is the least common multiple of
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 
.
We need to ensure that all the numbers share a common factor of
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What is the least common multiple of 
?
What is the least common multiple of
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 
.
We are inclined to multiply the numbers out however, if we divide both numbers by
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What is the least common multiple of 
?
What is the least common multiple of
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 
.
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What is the least common multiple of 
?
What is the least common multiple 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 
.
Both
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What is the least common multiple of 
?
What is the least common multiple of
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 
.
Both numbers are divisible by
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What is the least common multiple of 
?
What is the least common multiple of
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 
.
If we divide
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What is the least common multiple of 
?
What is the least common multiple of
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 numbers are divisible by
Compare your answer with the correct one above
What is the least common multiple of 
?
What is the least common multiple 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 
.
Both
Compare your answer with the correct one above
What is the least common multiple of 
?
What is the least common multiple of
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
Compare your answer with the correct one above
What is the least common multiple of 
?
What is the least common multiple of
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
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What is the least common multiple of the first six positive integers?
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 
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 
.
Let's divide the even numbers first. We will divide them by
Next, we have two
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Which can be a group of remainders when four consecutive integers are divided by 
?
Which can be a group of remainders when four consecutive integers are divided by
If you divide a number by 
, you cannot have a remainder of 
You can either have 
in that order.
If you divide a number by
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