Prime & Composite Numbers
Why do you only divide by prime numbers when you're trying to determine if a number is prime or composite?
You only divide by prime numbers since every number has prime factors.
So, if a number is not divisible by a prime number, it will not be divisible by any multiple of the prime number either.
Find the prime factorization of 105 using repeated division.
105 = 3 x 5 x 7
Fill in the blanks:
Scientific Notation is a way of writing a number as a ____________ between 1 and 10, _____________ by a power of 10.
Scientific Notation is a way of writing a number as a decimal between 1 and 10, multiplied by a power of 10.
What is a perfect square?
A perfect square is a number which has a whole number as its square root.
True or False:
When evaluating an expression, multiplication and division are evaluated in order from right to left.
False.
Multiplication and division are evaluated in order from left to right.
What is the divisibility rule for 6?
A number is divisible by 6 if it is divisible by both 2 and 3.
The prime factorization of 90 and 72 are shown below:
90 = 2 x 3 x 3 x 5
72 = 2 x 2 x 2 x 3 x 3
What is the GCF of 90 and 72?
The GCF of 90 and 72 is 18.
Write the following number in scientific notation:
45 010 000
45 010 000 = 4.501 X 10^7
What is the square root of 441?
You must first find the prime factorization of the number, then use it to solve.
The square root of 441 is 21.
True or False:
In an expression, square roots are to be treated like exponents (i.e. you must evaluate brackets before you evaluate square roots).
False.
Square roots are to be treated like brackets.
Is 79 a prime or composite number? Show your work!
79 is a prime number.
To determine this, keep dividing 79 by prime numbers.
Half of 79 is 39.5, so you can stop dividing once you've divided 79 by 41 (because 41 x 2 = 82, which is greater than 79; 2 is the smallest prime number that you can multiply a number by).
List all of the common factors of 90 and 72.
The common factors of 90 and 72 are:
2, 3, 6, 9, and 18
Write 43 091 203 in expanded form.
43 091 203 in expanded form is
4 x 10^7 + 3 x 10^6 + 9 x 10^4 + 1 x 10^3 + 2 x 10^2 + 3 x 1
Is 324 is perfect square?
You must prove this using the prime factorization of 324.
The prime factorization of 324 is 2 x 2 x 3 x 3 x 3 x 3.
The prime factorization has an even number of 2's and an even number of 3's.
Therefore, 324 is a perfect square!
Evaluate the following expression using order of operations (i.e. BEDMAS):
sqrt(31 + 10 / 2) + 3 x 2.5
sqrt(31 + 10 / 2) + 3 x 2.5
= sqrt(31 + 5) + 3 x 2.5
= sqrt(36) + 3 x 2.5
= 6 + 3 x 2.5
= 6 + 7.5
= 13.5
How do you know that 7^3 is not a prime number?
7^3 = 7 x 7 x 7, which indicates that this number has more than two factors.
Any combination of these factors, when multiplied, is a factor of this number (e.g. 7 x 7 = 49, so 49 is a factor of this number).
This proves that the number is not prime.
Problem 2) from Lesson 1.3 Note:
Observe the following Venn Diagram:
Determine the value of A and B.
A = 2 x 3 x 3 x 5
A = 90
B = 3 x 5 x 5
B = 75
Write the following number in standard form. Show your work!
5 x 10^8 + 1 x 10^5 + 4 x 10^4 + 3 x 10^2
5 x 10^8 + 1 x 10^5 + 4 x 10^4 + 3 x 10^2
= 500 000 000 + 100 000 + 40 000 + 300
= 500 140 300
Find a whole number whose square root is between 11 and 12. Explain how you got your answer.
11^2 or 11 x 11 = 121
12^2 or 12 x 12 = 144
So, any number between 121 and 144 will have a square root that is between 11 and 12.
E.g. 122, 123, 124, ..., 143 all have square roots between 11 and 12.
Casey solved the following expression; her solution is shown below:
4.5 x sqrt(6.5^2 + 3 x 2.25)
= 4.5 + 6.5 + 3 x 2.25
= 11 + 3 x 2.25
= 14 x 2.25
= 31.5
Explain all of the errors that Casey made. Be specific (there are 4 errors)! Then, correctly evaluate this expression.
Errors made by Casey:
1) She only found the square root of 6.5^2
2) She changed the x sign (after 4.5) to a + sign
3) She added 4.5 to 6.5 before multiplying 3 by 2.25
4) She added 11 to 3 before multiplying 3 by 2.25
Correct solution:
4.5 x sqrt(6.5^2 + 3 x 2.25)
= 4.5 x sqrt(42.25 + 3 x 2.25)
= 4.5 x sqrt(42.25 + 6.75)
= 4.5 x sqrt(49)
= 4.5 x 7
= 31.5 (answer is the same, but Casey's steps were incorrect)