# SAT II Math II : Number Theory

## Example Questions

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### Example Question #1 : Number Theory

Express  as a base ten number.

Explanation:

Place values in the base two system are powers of two rather than powers of ten.

is equal to:

### Example Question #1 : Number Theory

Express 68 as a base two (binary) number.

Explanation:

68 can be expressed as the sum of distinct powers of two as follows:

Therefore, there are 1's in the seventh and third place from the right, and 0's everywhere else, so

### Example Question #1 : Number Theory

Express 73 as a binary (base two) number.

Explanation:

73 can be uniquely expressed as the sum of distinct powers of two as follows:

Therefore, there is a "1" in the seventh, fourth, and first positions from the right, and a "0" everywhere else. So,

### Example Question #2 : Number Theory

Which of the following numbers is an imaginary number?

Explanation:

By definition, an imaginary number is a number with the term "i" in it. For this problem 7i is the only answer with the imaginary part.

### Example Question #1 : Irrational Numbers

Define an operation  on the set of real numbers as follows:

For any two real numbers ,

.

Evaluate .

Explanation:

Substitute  in the expression:

### Example Question #1 : Number Theory

The above represents a Venn diagram. The universal set  is the set of all positive integers.

Let  be the set of all multiples of 2; let  be the set of all multiples of 3; let  be the set of all multiples of 5.

As you can see, the three sets divide the universal set into eight regions. Suppose each positive integer was placed in the correct region. Which of the following numbers would be in the same region as 873?

Explanation:

From the last digit, it can be immediately determined that 873 is not a multple of 2 or 5; since , 873 is a multiple of 3. Therefore,

We are looking for an integer that is also in this set - that is, one that is also a multiple of 3 but not 2 or 5. From the last digits, we can immediately eliminate 366 and 368 as multiples of 2 and 365 as a multiple of 5. We test 367 and 369 to see which one is a multiple of 3:

369 is the correct choice.

### Example Question #3 : Number Theory

Which of the following sets is closed under multiplication?

Explanation:

A set is closed under multiplication if and only if the product of any two (not necessarily distinct) elements of that set is itself an element of that set.

This can easily be disproved in the case of three of these sets:

But closure can be proved to hold in the case of . Each number takes the form of  for some nonnegative integer . If we multiply two numbers in this form, we get

which is an element of , being of this form.

### Example Question #4 : Number Theory

What is the power set for the data set?

Explanation:

The power set is the set of all subsets that can be created from the original set.

For the set  , you can create the subsets:

This means that the power set is the set of all sets, so the power set is:

### Example Question #1 : Number Theory

Multiply:

None of the other responses is correct.

None of the other responses is correct.

Explanation:

This is the product of a complex number and its complex conjugate. They can be multiplied using the pattern

with

This is not among the given responses.

Multiply: