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Number Theory
Primes Name________________________ Period _____ A Prime Number is a whole number whose only factors are 1 and itself. To find all of the prime numbers between 1 and 100, complete the following exercise:
1. Cross out 1 by Shading in the box completely. 1 is neither prime nor composite. It has only 1 factor - itself. 2. Use a forward *Slash * to cross out all multiples of 2, starting with 4. 2 is the first prime number. 3. Use a backward Slash / to cross out all multiples of 3 starting with 6. 4. Multiples of 4 have been crossed out already when we did #2. 5. Draw a Square on all multiples of 5 starting with 10. 5 is prime. 6. Multiples of 6 should be X’d already from #2 and #3. 7. Circle all multiples of 7 starting with 14. 7 is prime. 8. Multiples of 8 were crossed out already when we did #2. 9. Multiples of 9 were crossed out already when we did #3. 10. Multiples of 10 were crossed out when we did #2 and #5. All of the remaining numbers are prime. How many prime numbers are left between 1 and 100? _____ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Answer: Use your chart for help. Is 51 prime? If not, what are its factors? ____________ Is 59 prime? If not, what are its factors? ____________ Is 87 prime? If not, what are its factors? ____________ Is 91 prime? If not, what are its factors? ____________
Number Theory
Divisibility Rules Name________________________ Period _____ There are some easy tricks you can use to determine if a number is divisible by 2, 3, 4, 5, 6, 8, 9 and 10. A number is divisible by: 2 - if it is even. 3 - if the sum of its digits is divisible by 3. 4 - if the number formed by the last 2 digits is divisible by 4. (ask me why this works) 5 - if the ones digit is 5 or 0. 6 - if it is divisible by 2 AND 3. (All even multiples of 3.) 7 - there is no good trick for 7. 8 - if the number formed by the last 3 digits is divisible by 8. (ask me why this works) 9 - if the sum of the digits is divisible by 9. 10 - if the last digit is a 0. 11: We will learn this trick next. It is more complicated. Practice: Write y es or n o in each blank. Determine whether 21,408 is divisible by: 2 - ____ 6 - ____ 3 - ____ 8 - ____ 4 - ____ 9 - ____ 5 - ____ 10 - ____ Determine whether 1,345,866 is divisible by: 2 - ____ 6 - ____ 3 - ____ 8 - ____ 4 - ____ 9 - ____ 5 - ____ 10 - ____ Determine whether 222,222,225 is divisible by: 2 - ____ 6 - ____ 3 - ____ 8 - ____ 4 - ____ 9 - ____ 5 - ____ 10 - ____
Number Theory
Divisibility Rule: Eleven The divisibility rule for 11 is seldom taught in regular classes. Practice: First, take a moment to multiply several numbers by 11:
504 1723 abcd
x11 x11 x
You should see some patterns with the digits. In the final example, the digits become: a a+b b+c c+d d If you add the alternating digits you get a a+b b+c c+d d the same result. To find out if a number is divisible by eleven: Sum the alternating digits. Subtract these two numbers. If the result is zero or is divisible by 11, the number is divisible by 11. Examples: Determine if each number is divisible by 11 without a calculator:
Practice: Determine if each number is divisible by 11 without a calculator:
Harder Practice: Solve each without a calculator.
1. What digit could fill-in the blank if 89_43 is divisible by 11?
2. What five-digit multiple of 11 consists entirely of 2s and 3s? 3. What is the largest five-digit multiple of 11? 4. What is the remainder when you divide 1,234,567 by 11?
Number Theory
Divisibility Practice Practice: Solve each using what you have learned about divisibility.
1. What digit could be used to fill in the blank and make the following number divisible by both 3 and 8? 45,2_ _____ 2. What is the smallest three-digit prime? _____ 3. How many multiples of 3 less than 1,000 use only the digits 2 and/or 4. _____ 4. 360 is divisible by both 8 and 9. How many integers less than 360 are also divisible by both 8 and 9? (hint: First find the smallest integer that is divisible by both 8 and 9.) _____ 5. A three-digit integer is divisible by 9. If I subtract the tens digit from the hundreds digit, I get the ones digit. What is the largest number that meets these conditions? _____ 6. There are two ways that the digits 1, 2, 3, and 4 be arranged to create a four-digit multiple of 8. Find them both. _____ _____ 7. Consecutive integers are placed in order to form a three-digit integer. The integer will ALWAYS be divisible by what prime number? _____ 8. For the number ABC, each distinct letter represents a different digit. If ABC, CAB, and BCA are all divisible by 6 and 9, find the value of ABC + CAB + BCA. _____ 9. What is the largest seven-digit number that contains each of the digits 1 through 7 and has the property that the sum of any two consecutive digits is a prime number? (source: MATHCOUNTS 2000 National Team Round) _____ Name________________________ Period _____
Number Theory
Quiz: Divisibility Solve:
8. The digits 5, 6, and 7 are arranged to create a three-digit number. Which of the following cannot be a factor of the number formed? (There may be more than one answer, list all that apply.) 2 3 4 5 6 8 9 8. _______ 9. Using two 5s and two 6s, it is possible to create four 4-digit numbers which are divisible by 11. What is the sum of these four numbers? 9. _______ 10. Each of the digits 0-9 is used exactly once to create a ten-digit integer. Find one of the many ten-digit numbers which uses each digit once and is divisible by 8, 9, 10, and 11. 10. ___________________ Name________________________ Period _____
Number Theory
Divisibility Combo Rules: Each player gets 6 cards, each with a single digit. Each hand, players’ goal is to find a number using at least 3 of their 6 digits which meets the divisibility requirements. Example: You hold 1,4,4,5,7 and 8. Goal: Divisibility by 3 and 5. Possible answer: 8,415 4,875 etc. Example: You hold 1,2,3,3,8 and 9. Goal: Divisibility by 3 and 8. Can you find a solution? Sometimes it will be impossible using the cards you have. After a player gets three combos, (s)he wins the round and everyone gets a new hand. Create a 3+ Digit Number that is Divisible by: 6 2 and 3 3 and 4 5 and 6 2 and 11 3 and 11 8 and 9 4 and 6 11 5 and 8 3 and 8 4 and 9 9 4 and 5 2 and 11 5 and 9 3 and 10 2 and 3 and 5
Number Theory
Testing for Primes Testing to see if a number is prime: If we want to know if 401 is prime, do we need to test to see if the following numbers are factors: 6? 7? 10? 13? 20? 23? Is 401 prime? How can you tell? To determine whether a number n is prime: Check for divisibility by primes < n starting from least to greatest. Think: You do not need to check composites like 6 and 14 because if 6 were a factor, 2 and 3 would be factors. If 14 were a factor, 7 would also be a factor. You do not need to check primes greater than the square root because they would be multiplied by a number less than the square root (which we would have already checked). Example: Is 181 prime? Check in your head: 2, 3, 5, and 11. Check 7 and 13 on paper if you need to. Make sure you understand why you do not need to check numbers like 4 and 17 as factors. Practice: Answer each (you wil need a calculator). If a number is composite, write two numbers whose product is the number.
1. Is 391 prime? 2. Is 287 prime? 3. Is 503 prime? 4. The number 13 is prime, and when its digits are reversed, 31 is also prime. In addition to 13 and 31, five other 2-digit primes satisfy this condition. What are they? Challenge: What is the smallest 4-digit prime?
- 17x23 2. 7x41 3. Prime 4. 11-11, 17-71, 37-73. C. 1001=7x11x13 1003=17x59 1007=19x53 1009 is :prime Notice that you do not need to check 7, 11, 13, 17, or 19 (for 1009) since they are factors of 1001, 1003, and 1007.
Number Theory
A Brief Intro to Exponents
One of the things that will come up frequently in this class which you all must be familiar with is the use of exponents. Base: The repeated factor in a power. In the expression n³ , n is the base. Exponent: Represents the number of times a factor is being multiplied. In the expression n³ , the ³ is the exponent. The expression 3
5 means that you multiply^5 ^5 ^5
The expression 5
x means that you multiply x x x x x
The expression 4
( ab ) means that you multiply^ ( ab )( ab )( ab )( ab )
Practice: Write-out without using exponents: 2 2 2 2 2 2
5
1. 5
6
n 3.
5
( 3 n ) 4.
3 4
2 a
Practice: Write using exponents. Ex. 5
1. a a a a 2. 3 3 3 5 5 7 3. ^
5
n n n
Practice: Evaluate. (solve) Ex.^232
5
1. 2
4
3 2
2 4
Number Theory
Fundamental Counting Principle Another important Fundamental in mathematics is The Fundamental Counting Principle: See Probability Unit for worksheet
Number Theory
Counting Factors One way to count the number of factors (divisors) that a number has is to list them. Examples: 40: 96: 196: When you list factors, list them in pairs and go from least to greatest. There is a nice relationship between the prime factorization of a number and the number of factors (divisors) that it has. Example: Look at the prime factorization of 40: 3 2 200 2 5 Every factor of 40 is a combination of 2s and 5s. 0 0 1 2 5 1 0 2 2 5 2 0 4 2 5 3 0 8 2 5 0 1 5 2 5 1 1 10 2 5 2 1 20 2 5 3 1 40 2 5 0 2 25 2 5 1 2 50 2 5 2 2 100 2 5 3 2 200 2 5 If you are only asked HOW MANY factors a number has, there is an easy shortcut that involves the prime factorization (maybe you can recognize what this shortcut is by looking at the example above): Example: How many factors does 56 have? First find the prime factorization. 3 1 56 2 7 Each factor can have either 0, 1, 2, or 3 twos and 0 or 1 seven. This means that 56 has 4 choices for the number of twos and two choices for the number of sevens in each of its factors, for a total of 4x2=8 factors. They are 1, 2, 4, 7, 8, 14, 28, and 56 This is a great trick and we will practice more with it.
Number Theory
Counting Factors Trickier factor counting: Examples:
1. How many odd factors does 240 have? Reasoning: First look at the prime factorization of 240: 4 1 1 240 2 3 5 Any number times 2 is even, so the odd factors are the ones that have no twos. I can only use the 3s and 5s for a total of 2x2=4 factors: 1, 3, 5, and 15. 2. How many perfect squares are factors of 360? Reasoning: Look again at the number’s prime factorization: 3 2 1 360 2 3 5 We need to find factors that have two twos, two threes, or both. 51 cannot be a factor of a perfect square. Our perfect squares are 2^0 =1, 2^2 =2x2=4, 3^2 =3x3=9, and 2^2 x3^2 =2x2x3x3=36. Finding the prime factorization of a number quickly is the key to MANY number theory problems. Practice: 1. How many perfect squares are factors of 400? 2. How many even factors does 210 have? 3. If a number n has 7 factors, how many factors does n^2 have? 4. The number n is a multiple of 7 and has five factors. How many factors does 3n have? Challenge: The number p is a multiple of 6 and has 9 factors. How many factors does 10 p have?
- 6: 1, 4, 16, 25, 100, 400 2. 8: 2, 6, 10, 14, 30, 42, 70, 210 3. 13 (consider n^6 and n^12 )
- The original number must be 7^4 so the new number 3x7^4 has 10 factors C. 24 (p=2x2x3x3)
Number Theory
Counting Factors Basics: How many factors (divisors) does each number have? (do not list them)
1. 480 2. 400 Basics: How many ODD factors (divisors) does each number have? (do not list them) 3. 900 4.^6 ,^600 Basics: How many EVEN factors (divisors) does each number have? (do not list them) 5. 450 6.^3 ,^200 Basics: How many Perfect Square factors (divisors) does each number have? List them. 7. (^) 210 8.^1 ,^296 Apply what you know: 9. How many two-digit numbers have EXACTLY three factors? 10. What is the smallest positive integer that has 18 factors? Name________________________ Period _____
Number Theory
Practice Quiz: Factors & Divis. Solve:
1. How many of the following integers are factors of 888? 2, 3, 4, 5, 6, 8, 9, 10. 1. _______ 2. What number could fill-in the blank to make 8_,045 divisible by 11? 2. _______ 3. What is the smallest prime number that is greater than 150? 3. _______ 4. What is the Prime Factorization of 264 (written with exponents)? 4. _____________________ 5. How many factors does 600 have? 5. _______ 6. The digits of a 5-digit positive integer are 1s, 2s, and 3s with at least one of each. What is the smallest such integer that is divisible by both 8 and 9? 6. _______ 7. What is the smallest positive integer that has exactly 10 factors? 7. _______ Name________________________ Period _____
Number Theory
Practice Quiz: Factors & Divis. Solve:
8. Distinct positive integers a and b have 5 and 6 factors respectively. What is the smallest possible product ab if a and b do not have any factors in common greater than 1? 8. _______ 9. How many even factors does 990 have? 9. _______ 10. How many 3-digit integers have exactly 3 factors? 10. _______ Name________________________ Period _____
