SAT Math : Arithmetic

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #52 : Factors / Multiples

If k and m are positive integers, let m be equal to the reaminder when k is divided by m. For example, 3 # 2 = 1. All of the following are true EXCEPT:

Possible Answers:

k = m

0 # (m) = 0

(km) # k = m

m = 0

(k)# k = 0

Correct answer:

(km) # k = m

Explanation:

Let's look at each statement separately and see if it must be true.

First, let's consider 0 # (m) = 0.

This statement essentially says that if we divide 0 by the sum of k and m, the remainder should be 0. In general, when 0 is divided by an integer (other than itself), the result is zero with a remainder of zero. For example, if 0 is divided by 2, then 2 will go into zero exactly zero times with a remainder of zero. Because the sum of k and m is an integer, this means that zero divided by (m) will also have a remainder of zero. Thus, 0 # (m) = 0 is true.

Next, let's examine m = 0. This statement says that if we divide m by itself, we should get a remainder of zero. This is true, because any integer divided by itself is equal to 1, with a remainder of zero. Remember that the result of using the # symbol only gives us the remainder. Therefore m # m = 0, not 1. 

Let's now look at k # k = m # m.

As we established previously, m # m = 0. Similarly, k = 0, because any integer divided by itself produces a remainder of zero. Because m # m and k # k both equal zero, they are also equal to each other. Thus, k # k = m # m is indeed true.

Next, let's consider (k2) # k = 0. This says that if we square an integer and then divide the result by the integer, we should get a remainder of zero. This is true, because an integer is always a factor of its square. In other words, the square of an integer is always divisible by the integer. And if one number is divisible by another, then the resulting quotient has a remainder of zero. For example (52) # 5 = 0, because when we divide 25 by 5, we get 5 with a remainder of zero. Again, remember that we are only concerned with the remainder. 

By the process of elmination, the answer is (km) # k = m. Let's see why this statement must be false. The product of k and m has factors k and m. For example, the product of 4 and 3, which equals 12, has the factors 4 and 3. When a number is divided by one of its factor, the resulting remainder must be zero. For example (4(3)) # 4 = 12 # 4 = 0, because the remainder when 12 is divided by 4 is equal to zero. Thus (km) # k = 0, not m

The answer is (km) # k = m.

Example Question #61 : Factors / Multiples

If    180 = 2^{a}3^{b}5^{c}7^{d}, where a,b,c,d are all positive integers, what is a+b+c+d?

Possible Answers:

4

5

3

7

6

Correct answer:

5

Explanation:

We will essentially have to represent 180 as a product of prime factors, because 2, 3, 5, and 7 are all prime numbers. The easiest way to do this will be to find the prime factorization of 180.

180 = 18(10)= (9)(2)(10) = (3)(3)(2)(10)=(3)(3)(2)(2)(5) = 2^{2}3^{2}5^{1}. Because 7 is not a factor of 180, we can mutiply the prime factorization of 180 by 7^{0} (which equals 1) in order to get 7 into our prime factorization.

180= 2^23^25^17^0= 2^a3^b5^c7^d

In order for 2^23^25^17^0 to equal 2^a3^b5^c7^d, the exponents of each base must match. This means that a = 2, b = 2, c = 1, and d = 0. The sum of a, b, c, and d is 5.

The answer is 5.

Example Question #11 : How To Factor A Number

What is the product of the distinct prime factors of 24?

Possible Answers:

\dpi{100} \small 8

\dpi{100} \small 24

\dpi{100} \small 9

\dpi{100} \small 5

\dpi{100} \small 6

Correct answer:

\dpi{100} \small 6

Explanation:

The prime factorization of 24 is (2)(2)(2)(3).  The distinct primes are 2 and 3, the product of which is 6.

Example Question #12 : How To Factor A Number

How many prime factors does \dpi{100} \small 2^{3}-1 have?

Possible Answers:

\dpi{100} \small 2

\dpi{100} \small 0

\dpi{100} \small 1

\dpi{100} \small 5

\dpi{100} \small 3

Correct answer:

\dpi{100} \small 1

Explanation:

\dpi{100} \small 2^{3}-1=8-1=7

Since 7 is prime, its only prime factor is itself.

Example Question #301 : Arithmetic

What is the smallest positive multiple of 12?

Possible Answers:

\dpi{100} \small 12

\dpi{100} \small 0

\dpi{100} \small 2

\dpi{100} \small 24

\dpi{100} \small 6

Correct answer:

\dpi{100} \small 12

Explanation:

Multiples of 12 are found by multiplying 12 by a whole number.  Some examples include:

\dpi{100} \small 12(-2)=-24

\dpi{100} \small 12(0)=0

\dpi{100} \small 12(1)=12

Clearly, the smallest positive value obtainable is 12.  Do not confuse the term multiple with the term factor!

Example Question #31 : Factors / Multiples

How many prime factors of 210 are greater than 2?

Possible Answers:

one

three

two

four

five

Correct answer:

three

Explanation:

Begin by identifying the prime factors of 210. This can be done easily using a factoring tree (see image).

Vt_p2

 

 The prime factors of 210 are 2, 3, 5 and 7. Of these factors, three of them are greater than 2. 

Example Question #32 : Factors / Multiples

How many integers between 50 and 100 are divisible by 9?

Possible Answers:

8

7

6

9

5

Correct answer:

6

Explanation:

The smallest multiple of 9 within the given range is \inline \dpi{200} \tiny 54 = 9 \times 6.

The largest multiple of 9 within the given range is \dpi{100} {99=9 \times 11}.

Counting the numbers from 6 to 11, inclusive, yields 6.

Example Question #14 : Other Factors / Multiples

  is the set of all positive multiples of , and  is the set of all squares of integers. Which of the following numbers belongs to both sets?

Possible Answers:

Correct answer:

Explanation:

 is the only choice that is both a multiple of  and a perfect square. 

Example Question #761 : Arithmetic

The sum of three consecutive even integers is 108.  What is the largest number?

Possible Answers:

42

36

34

38

40

Correct answer:

38

Explanation:

Three consecutive even integers can be represented by x, x+2, x+4.  The sum is 3x+6, which is equal to 108.  Thus, 3x+6=108.  Solving for x yields x=34.  However, the question asks for the largest number, which is x+4 or 38.  Please make sure to answer what the question asks for!

You could have also plugged in the answer choices.  If you plugged in 38 as the largest number, then the previous even integer would be 36 and the next previous even integer 34.  The sum of 34, 36, and 38 yields 108.

Example Question #1 : Consecutive Integers

The sum of three consecutive even integers equals 72. What is the product of these integers? 

Possible Answers:

17472

13800

12144

10560

13728

Correct answer:

13728

Explanation:

Let us call x the smallest integer. Because the next two numbers are consecutive even integers, we can call represent them as x + 2 and x + 4. We are told the sum of x, x+2, and x+4 is equal to 72. 

x + (x + 2) + (x + 4) = 72

3x + 6 = 72

3x = 66

x = 22.

This means that the integers are 22, 24, and 26. The question asks us for the product of these numbers, which is 22(24)(26) = 13728. 

The answer is 13728.

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