How to find the greatest common factor - PSAT Math
Card 0 of 42
; 
, 
, and 
are distinct integers.
Which of the following is a possible value of 
?
Which of the following is a possible value of
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We need to find ways to factor 18 such that the three factors are different, and then find the sum of those factors in each case.
18 can be factored as the product of three integers in four ways:
I) 
II) 
III) 
IV) 
Disregard I and IV since each repeats a factor.
In (II), the sum of the factors is 
; in (III) the sum is 
. Of the five choices, only 12 is possible.
We need to find ways to factor 18 such that the three factors are different, and then find the sum of those factors in each case.
18 can be factored as the product of three integers in four ways:
I)
II)
III)
IV)
Disregard I and IV since each repeats a factor.
In (II), the sum of the factors is
Suppose that 
is an integer such that 
is ten greater than 
. What is the value of 
?
Suppose that
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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.
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.
If 
is divisible by 2, 3 and 15, which of the following is also divisible by these numbers?
If
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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.
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.
, 
, and 
are positive two-digit integers.
The greatest common divisor of 
and 
is 10.
The greatest common divisor of 
and 
is 9.
The greatest common divisor of 
and 
is 8.
If 
is an integer, which of the following could it be equal to?
The greatest common divisor of
The greatest common divisor of
The greatest common divisor of
If
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The greatest common divisor of 
and 
is 10. This means that the prime factorizations of 
and 
must both contain a 2 and a 5.
The greatest common divisor of 
and 
is 9. This means that the prime factorizations of 
and 
must both contain two 3's.
The greatest common divisor of 
and 
is 8. This means that the prime factorizations of 
and 
must both contain three 2's.
Thus:
We substitute these equalities into the given expression and simplify.
Since 
and 
are two-digit integers (equal to 
and 
respectively), we must have 
and 
. Any other factor values for 
or 
will produce three-digit integers (or greater).
is equal to 
, so 
could be either 1 or 2.
Therefore:
or
The greatest common divisor of
The greatest common divisor of
The greatest common divisor of
Thus:
We substitute these equalities into the given expression and simplify.
Since
Therefore:
or
.
, 
, and 
are integers; they may or may not be distinct.
Which of the following cannot be equal to 
?
Which of the following cannot be equal to
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We look for ways to write 45 as the product of three integers, then we find the sum of the integers in each situation. They are:
Sum: 
Sum: 
Sum: 
Sum: 
Of the five choices, only 33 is not a possible sum of the factors. This is the correct choice.
We look for ways to write 45 as the product of three integers, then we find the sum of the integers in each situation. They are:
Sum:
Sum:
Sum:
Sum:
Of the five choices, only 33 is not a possible sum of the factors. This is the correct choice.
What is the greatest common factor of 
and 
?
What is the greatest common factor of
Tap to see back →
The greatest common factor is the largest factor that both numbers share. Each number has many factors. The factors for 72 are as follows:
Starting from the largest factor, 72, we can see that it is also a factor of 144
.
Therefore, 72 is the greatest common factor.
The greatest common factor is the largest factor that both numbers share. Each number has many factors. The factors for 72 are as follows:
Starting from the largest factor, 72, we can see that it is also a factor of 144
Therefore, 72 is the greatest common factor.
; , , and are distinct integers.
Which of the following is a possible value of ?
Which of the following is a possible value of
Tap to see back →
We need to find ways to factor 18 such that the three factors are different, and then find the sum of those factors in each case.
18 can be factored as the product of three integers in four ways:
I)
II)
III)
IV)
Disregard I and IV since each repeats a factor.
In (II), the sum of the factors is ; in (III) the sum is . Of the five choices, only 12 is possible.
We need to find ways to factor 18 such that the three factors are different, and then find the sum of those factors in each case.
18 can be factored as the product of three integers in four ways:
I)
II)
III)
IV)
Disregard I and IV since each repeats a factor.
In (II), the sum of the factors is
Suppose that is an integer such that is ten greater than . What is the value of ?
Suppose that
Tap to see back →
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.
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.
If is divisible by 2, 3 and 15, which of the following is also divisible by these numbers?
If
Tap to see back →
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.
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.
, , and are positive two-digit integers.
The greatest common divisor of and is 10.
The greatest common divisor of and is 9.
The greatest common divisor of and is 8.
If is an integer, which of the following could it be equal to?
The greatest common divisor of
The greatest common divisor of
The greatest common divisor of
If
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The greatest common divisor of and is 10. This means that the prime factorizations of and must both contain a 2 and a 5.
The greatest common divisor of and is 9. This means that the prime factorizations of and must both contain two 3's.
The greatest common divisor of and is 8. This means that the prime factorizations of and must both contain three 2's.
Thus:
We substitute these equalities into the given expression and simplify.
Since and are two-digit integers (equal to and respectively), we must have and . Any other factor values for or will produce three-digit integers (or greater).
is equal to , so could be either 1 or 2.
Therefore:
or
The greatest common divisor of
The greatest common divisor of
The greatest common divisor of
Thus:
We substitute these equalities into the given expression and simplify.
Since
Therefore:
or
.
, , and are integers; they may or may not be distinct.
Which of the following cannot be equal to ?
Which of the following cannot be equal to
Tap to see back →
We look for ways to write 45 as the product of three integers, then we find the sum of the integers in each situation. They are:
Sum:
Sum:
Sum:
Sum:
Of the five choices, only 33 is not a possible sum of the factors. This is the correct choice.
We look for ways to write 45 as the product of three integers, then we find the sum of the integers in each situation. They are:
Sum:
Sum:
Sum:
Sum:
Of the five choices, only 33 is not a possible sum of the factors. This is the correct choice.
What is the greatest common factor of and ?
What is the greatest common factor of
Tap to see back →
The greatest common factor is the largest factor that both numbers share. Each number has many factors. The factors for 72 are as follows:
Starting from the largest factor, 72, we can see that it is also a factor of 144
.
Therefore, 72 is the greatest common factor.
The greatest common factor is the largest factor that both numbers share. Each number has many factors. The factors for 72 are as follows:
Starting from the largest factor, 72, we can see that it is also a factor of 144
Therefore, 72 is the greatest common factor.
; , , and are distinct integers.
Which of the following is a possible value of ?
Which of the following is a possible value of
Tap to see back →
We need to find ways to factor 18 such that the three factors are different, and then find the sum of those factors in each case.
18 can be factored as the product of three integers in four ways:
I)
II)
III)
IV)
Disregard I and IV since each repeats a factor.
In (II), the sum of the factors is ; in (III) the sum is . Of the five choices, only 12 is possible.
We need to find ways to factor 18 such that the three factors are different, and then find the sum of those factors in each case.
18 can be factored as the product of three integers in four ways:
I)
II)
III)
IV)
Disregard I and IV since each repeats a factor.
In (II), the sum of the factors is
Suppose that is an integer such that is ten greater than . What is the value of ?
Suppose that
Tap to see back →
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.
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.
If is divisible by 2, 3 and 15, which of the following is also divisible by these numbers?
If
Tap to see back →
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.
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.
, , and are positive two-digit integers.
The greatest common divisor of and is 10.
The greatest common divisor of and is 9.
The greatest common divisor of and is 8.
If is an integer, which of the following could it be equal to?
The greatest common divisor of
The greatest common divisor of
The greatest common divisor of
If
Tap to see back →
The greatest common divisor of and is 10. This means that the prime factorizations of and must both contain a 2 and a 5.
The greatest common divisor of and is 9. This means that the prime factorizations of and must both contain two 3's.
The greatest common divisor of and is 8. This means that the prime factorizations of and must both contain three 2's.
Thus:
We substitute these equalities into the given expression and simplify.
Since and are two-digit integers (equal to and respectively), we must have and . Any other factor values for or will produce three-digit integers (or greater).
is equal to , so could be either 1 or 2.
Therefore:
or
The greatest common divisor of
The greatest common divisor of
The greatest common divisor of
Thus:
We substitute these equalities into the given expression and simplify.
Since
Therefore:
or
.
, , and are integers; they may or may not be distinct.
Which of the following cannot be equal to ?
Which of the following cannot be equal to
Tap to see back →
We look for ways to write 45 as the product of three integers, then we find the sum of the integers in each situation. They are:
Sum:
Sum:
Sum:
Sum:
Of the five choices, only 33 is not a possible sum of the factors. This is the correct choice.
We look for ways to write 45 as the product of three integers, then we find the sum of the integers in each situation. They are:
Sum:
Sum:
Sum:
Sum:
Of the five choices, only 33 is not a possible sum of the factors. This is the correct choice.
What is the greatest common factor of and ?
What is the greatest common factor of
Tap to see back →
The greatest common factor is the largest factor that both numbers share. Each number has many factors. The factors for 72 are as follows:
Starting from the largest factor, 72, we can see that it is also a factor of 144
.
Therefore, 72 is the greatest common factor.
The greatest common factor is the largest factor that both numbers share. Each number has many factors. The factors for 72 are as follows:
Starting from the largest factor, 72, we can see that it is also a factor of 144
Therefore, 72 is the greatest common factor.
; , , and are distinct integers.
Which of the following is a possible value of ?
Which of the following is a possible value of
Tap to see back →
We need to find ways to factor 18 such that the three factors are different, and then find the sum of those factors in each case.
18 can be factored as the product of three integers in four ways:
I)
II)
III)
IV)
Disregard I and IV since each repeats a factor.
In (II), the sum of the factors is ; in (III) the sum is . Of the five choices, only 12 is possible.
We need to find ways to factor 18 such that the three factors are different, and then find the sum of those factors in each case.
18 can be factored as the product of three integers in four ways:
I)
II)
III)
IV)
Disregard I and IV since each repeats a factor.
In (II), the sum of the factors is
Suppose that is an integer such that is ten greater than . What is the value of ?
Suppose that
Tap to see back →
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.
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.