Number Theory

Prime numbers

100

A prime number is...

A number which has no non-trivial factors (meaning a number that can only be divided by itself and 1)

200

Is 97 a prime number? Explain/show work to justify your answer

7x7 = 49

11x11 = 121 (too large)

2 -> No

3-> No

5 -> No

7 -> No

So YES, 97 is a prime number.

300

We need only to divide a potential prime number n by prime numbers which are...

Less than or equal to the square root of n.

For example:

Is 133 prime?

We check up until prime number 11 because 11x11=121. 13x13=169

400

Is 607 a prime number? Explain/show work to justify your answer.

23 x 23 = 529

29 x 29 = 841 (too large)

2 -> No

3 -> No

5 -> No

7 -> No

11 -> No

13 -> No

19 -> No

23 -> No

So YES, 607 is prime

500

Can we ever "run out" of prime numbers? Explain your reasoning

No, we can not run out of prime numbers. By creating a list, multiplying all primes together, and adding 1, we receive a new number that which is either:

(a) also prime

(b) divisible by a prime number not on our list

either way, our list of primes will be incomplete.

Prime factorization

100

Define the Fundamental Theorem of Arithmetic

Every number can be written as a product of prime numbers in exactly one way

200

True or false, the prime factorization of 68 is 2¹ x 34¹

False. It is 2² x 17¹

300

Complete the prime factorization for 120

2³ x 3¹ x 5¹

400

Complete the prime factorization for 322

2¹ x 7¹ x 23¹

500

Complete the prime factorization for 3542

2¹ x 7¹ x 11¹ x 23¹

Number sets

100

The set of all irrational real numbers includes...

the set of all non-repeating decimals (ie, 𝝅 )

200

The set of all rational real numbers includes...

The set of all fractions Q aka the set of all repeating decimals

300

Name three types of numbers and describe them in terms of sets

natural numbers N = {1, 2, 3, ...}

fractions Q = {p/q : p, q ∈ N, q ≠ 0}

integers Z = {... -2, -1, 0, 1, 2, ...}

400

define "infinite decimal expansion" and provide an example to justify your definition

infinite decimal expansion means that any decimal can be defined to any FINITE number of decimal places (in order to use it algebraically).

Ex: pi is not 3.14159..., it is "pi to one hundred decimal places"

500

Describe the set of all non-repeating decimals in symbols using principles of set theory.

Hint: Consider the diagram of real numbers R from the notes

Non-repeating decimals = R \ Q

Fractions -> Decimals

100

Complete the sentence:

When turning a fraction to a decimal, we ___________

use long division

200

Name the potential remainders we may get if we are dividing 247 by 8.

{0, 1, 2, 3, 4, 5, 6, 7}

aka {0, 1, 2, ... q-1}

300

Convert 4/15 into a repeating decimal

0.26666...

400

Convert 8/11 into a repeating decimal

0.727272...

500

Convert 15/77 into a repeating decimal

0.1948905...

Decimals -> Fractions

100

Complete the sentence:

When turning a repeating decimal into a fraction, we...

multiply by tens twice, then subtract.

200

If we are turning 2.47777... into a fraction, we multiply first by....and second by....

first by 100. 2.4777 becomes 247.777

second by 10. 2.4777 becomes 24.777

300

Convert 5.4444... into a fraction

10x = 54.44

1x = 5.44

9x = 49

x = 49/9

400

Convert 18.435656...into a fraction

920861/49950

500

Convert 431.0639843984... into a fraction

71836813/166650