Number Theory Concepts

Modeling the mean of a set of numbers

1. Describe with pictures and/or words how you could model …

a) the mean of 3, 11, and 10

b) the mean of 3, 8, 5, and 4

c) the mean of 3, 7, 4, and 2

d) the mean of 2, 10, and 3

2. Do not disturb the results from the last problem (1d) and determine a fourth number you would need to place with the three numbers to raise the mean of the four numbers to 9.

3. If your average on three exams is 87, what will you have to make on the fourth exam to raise your average to 90?

4. If the mean of five numbers is 16, what is the sum of the five numbers? _____

5. Suppose the mean annual salary of 8 workers is $24,500. Their supervisor’s salary is $47,000. What is the mean salary of the 9 employees?

Modeling number concepts

1. Use a model to explain why 7 is an odd number and 8 is an even number?

2. In class, square tiles were used to model even and odd numbers. Could we have used circular chips? Why or why not?

3. Using square tiles to represent any number as a rectangular array, describe how prime numbers were different from composite numbers? How were square numbers different from other composite numbers?

4. Circle all the prime numbers between 60 and 70. 61 62 63 64 65 66 67 68 69

5. Circle all the prime numbers between 90 and 100. 91 92 93 94 95 96 97 98 99

6. Circle all the square numbers between 75 and 125. 75 76 77 78 79 80 81 82 83 84

85 86 87 88 89 90 91 92 93 94

95 96 97 98 99 100 101 102 103 104 etc

7. Circle all the square numbers between 150 and 200.

8. On a Hundred Chart, circle multiples of 3. Draw a box around each multiple of 4. Common multiples of 3 and 4 are both circles and boxed. List the common multiples of 3 and 4. ________ What is the least common multiple of 3 and 4? LCM(3,4) = _____

9. Repeat the previous exercise for 3 and 6. For 4 and 6. For 12 and 15.

10. On a Hundred Chart, circle 12. Circle all the divisors of 12. Draw a box around 16. Common divisors of 12 and 16 are both circled and boxed. List the common divisors of 12 and 16. _________ What is the greatest common divisor of 12 and 16? GCD(12, 16) = ________

11. Repeat the previous exercise for 16 and 24. For 15 and 25.

Prime Numbers and Factors

1. What is a prime number?

2. What is the smallest odd prime number?

3. How does 2 differ from all the other prime numbers?

4. What are twin primes? List five pairs of twin primes.

5. Mark the prime numbers that are factors of 12.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

6. Mark the prime numbers that are factors of 16.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

7. Mark the prime numbers that are factors of 18.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

8. Mark the prime numbers that are factors of 20.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

9. Use a factor tree to factor 24 into prime factors. 24 =

10. Use a factor tree to find the prime factorization of 36. 36 =

11. Write 40 as the product of prime factors. 40 =

Consecutive counting numbers

1. Aside from adding the numbers, what is a quick way of finding the sum of …

a) 5 consecutive counting numbers?

b) 7 consecutive counting numbers?

c) 4 consecutive counting numbers?

d) 8 consecutive counting numbers?