SAT Math : Arithmetic

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #1 : Factors / Multiples

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 

Example Question #71 : Integers

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 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 

Example Question #11 : 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 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 

Example Question #1591 : Sat Mathematics

What is the least common multiple of the first six positive integers?

Possible Answers:

Correct answer:

Explanation:

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 

Example Question #11 : Least Common Multiple

Which can be a group of remainders when four consecutive integers are divided by ?

Possible Answers:

Correct answer:

Explanation:

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

Example Question #1591 : Sat Mathematics

If a, b, and c are positive integers such that 4a = 6b = 11c, then what is the smallest possible value of c?

Possible Answers:

67

33

132

11

121

Correct answer:

67

Explanation:

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.

Example Question #1 : Factors / Multiples

If  is divisible by 2, 3 and 15, which of the following is also divisible by these numbers?

Possible Answers:

Correct answer:

Explanation:

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.

Example Question #1 : Factors / Multiples

Suppose that  is an integer such that  is ten greater than . What is the value of ?

Possible Answers:

Correct answer:

Explanation:

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.

Example Question #2 : Factors / Multiples

, , 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?

Possible Answers:

Correct answer:

Explanation:

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 

Example Question #1 : How To Find The Greatest Common Factor

What's the greatest common factor of 6 and 8?

Possible Answers:

Correct answer:

Explanation:

Greatest common factor is a common factor shared by two or more numbers. Both numbers are even, so let's divide both numbers by two. We get . These are prime numbers (factors of one and itsef) in which we are done. Anytime we have two prime numbers or one prime and one composite number, we are finished. So the greatest common factor is .

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