C6 Working with Prime Numbers

Question 1

the 15 prime numbers under 50

Answer 1

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47

Question 2

11 prime numbers from 50 to 102

Answer 2

53 59 61 67 71

73 79 83 89 97 101

Question 3

prime factorization

Answer 3

the prime factorization of a number is the unique product of the prime numbers that results in the given number.

Question 4

prime factorization methods

Answer 4

1. The tree
2. Upside down division

2|120

2|60

2|30

3|15

5

= 2³ x 3 x 5

Question 5

rules of divisibility for numbers 2 thru 12

Answer 5

2: the number ends in 0, 2, 4, 6, 8
5: the number ends in 0 or 5
10: the numbers ends in 0
4: the last two digits form a number divisible by 4
8: the last three digits form a number divisible by 8
3: the sum of the digits is divisible by three
9: the sum of the digits is divisible by 9
11: the difference between the sum of alternating digits is divisible by 11 or is 0.
6: the number is divisible by both 2 and 3 (use both rules)
12: the number is divisible by both 3 and 4 (use both rules

Question 6

prime factorization to reduce fractions: what are the steps?

Answer 6

1. find the prime factorization of the numerator
2. find the prime factorization of the denominator
3. write the fraction with the prime factorization
4. cross out any factors in common

100/243 = 2³ x 5²/ 3⁵ has no common factors so it is reduced all the way. You can write the fraction in either form

Question 7

what is the subtle difference when dividing fractions with exponents and reducing fractions using prime factorization with exponents?

Answer 7

dividing fractions: with exponents you subtract the exponent in the numerator from the denominator

reducing fractions using prime factorization: you cross out the exponents so essentally you find the difference in the two

48x³z / 84xz⁴ = 4x²/ 7z³

Question 8

pulling out the factors and leaving the rest: what are the steps to simplifying expressions by finding the GCF and pulling it out of a list of terms?

12x²y⁴ + 16xy³ - 20x³y²

Answer 8

12x²y⁴ + 16xy³ - 20x³y²

1. Determine any common numerical factors.

[each term has a coefficient that is divisible by a power of 2, which is 2² = 4]

2. Determine any common variable factors.

[each term has x and y factors]

3. Write the prime factorizations of each term.

= 12x²y⁴ = 2² * 3 * x²y⁴

= 16xy³ = 2⁴ * xy³

= -20x³y² = 2² * 5 * x³y²

4. Find the GCF.

=4xy²

5. Divide each term by the GCF.

= 12x²y⁴/4xy² = 3xy²

= 16xy³/4xy² = 4y

= -20x³y²/4xy² = -5x²

6. Write the result as a product of the GCF and the results of the division.

=4xy²(3xy² + 4y - 5x²)