Math Exam 2

prime

A______________ number only has two divisors: 1 and itself

equation

When one makes a statement which sets two expressions equal to each other, on has created an___________.

quotient

In the equation 15/3 =5 the number 5 is called the_____________

divisor

After estimating the quotient and calculating the remainder in long division, it is necessary to check that 0 is less than or equal to the remainder less than or equal to____________

expression

A(n) ______________ is a combination of numbers, parenthesis, operations, and letters which represent a number.

1. Base

2. exponent

When writing a number such as b”=y, b is called the______________ and m is called the _____________.

proof

A _______________ is a chain of reasoning that explains why the fact is a consequence of previously known facts.

1. under

2. over

3. over

4. under

5. under

a. Indicating the weight that an elevator can hold ________________

b. Estimating the weight of passengers on an elevator, compared to its capacity ___________

c. Estimating the amount of food needed for a banquet ____________________

d. Determining the safe driving speed on a street ____________

e. Estimating how many miles you can drive with 1/4 a tank of gas in your car ____________

False (is 2 is one prime number then the sum is odd)

Whenever p and q are prime numbers, p+q is even. T/F

True

If a number is divisible by 40, then it is divisible by both 4 and 10. T/F

False ex: 20

If a number is divisible by both 4 and 10, then it is divisible by 40. T/F

False—— divisibility by 4 is determined by last two dis

If a number is divisible by 4, then the sum of all of its digits are divisible by 4. T/F

False: product of odds is odd

When you multiply two odd numbers together, you always get an even number

a. 10^4

b. 2²0

Write the following numbers as an exponent.

a. 10,000

b. 1024

Simplify. Leave your answer in prime factors.

Find the prime factorization for 4653.

2,3,4,11

Circle the number below that divide the number 1,702,932

2 3 4 5 8 9 11

Find a 5 digit number ending with the digits 7263 which is divisible by 11

What is the largest prime number you have to check in order to determine if 257 is prime or composite?

a. (40+1)²

b. 600

c. 169,000

d. 2800/ 70= 40

20. Use mental math to compute the following. Write your answer in a way that clearly shows the steps involved in solving the problem mentally.

a. 41²=

b. 420,000

c. 130 × 1300

d. Give a simple estimate for 2913 / 73

Use Algebra to give a complete teacher’s solution to solve: A pineapple costs 3 times as much as a mango. Lisa bought 3 mangoes and 4 pineapples for $30. Determine the cost of one mango.

Every 28 days

At a summer camp, chocolate milk is served every other day, corn is served every 4 days and pizza every 7 days. Today all three were served. What is the smallest number of days until all three are served again?

Use the Euclidean algorithm to find the greatest common divisor of 210 and 560. Show all your work.

Prove that a 3-digit number is divisible by 9 if and only if the sum of its digits is divisible by 9.

MATHEMATICALLY explain why 3/0 is undefined

Solve the following problem. Explain what each step of the division algorithm means in terms of the problem. David has 74 wheels. If he uses 4 wheels to make a toy car, how many toy cars can he make? How many wheels will be left over?

Use prime factorization to find the GCF and LCM of 28 and 63.

1280 / 40 = 320

Show how to compute the following mentally (eg. for 98 + 24 write = 100 + 22 = 122) Estimate using compatible numbers: 1270 ÷ 39 ≈

1380+200=1580

Overestimate the following. Show all steps. 1373 + 199

a. 4000 / <0=80

b. 3200/ 40=80

Estimate the following by reducing the problem to 1-digit multiplication facts. Be sure to show both your reasoning and your answer.

(a) 4123 ÷ 51

(b) 3187 ÷ 39

Show how to use compensation to estimate the following.

(a) 29, 737 + 42, 196 ≈ + = .

(b) 49, 432 − 24, 567 ≈ − = .

(c) 59 × 92 ≈ × = . (d) 5587 ÷ 79 ≈ ÷ = .

In the following situations, is it important to find an overestimate or underestimate? (Write “over” or “under”.)

(a) Indicating the weight that an elevator can hold .

(b) Estimating the weight of passengers on an elevator, compared to its capacity .

(c) Estimating the amount of food needed for a banquet .

(d) Determining the safe driving speed on a street .

(e) Estimating how many miles you can drive with 1/4 a tank of gas in your car

a. 55 × 20=1100

b. 60 × 25=1500

(a) Give an underestimate for 57 × 23.

(b) Give an overestimate for 57 × 23.

divisor

After estimating the quotient and calculating the remainder in long division, it is necessary to check that 0 ≤ remainder < .

Here are four student solutions to long-division problems. Correct them as a teacher would. Mark correct solutions with a checkmark. Otherwise, circle and explain the error

Give a Teacher’s Solution for the following problem: Ms. Contador baked 840 cookies and put them in boxes. Each box contained 24 cookies. If she sold all the boxes for $4 each, how much money did she receive?

a set of steps

Define algorithm

Compute the expression below without using the algorithms, but using the mental math strategies we have learned instead. Write all the steps you have taken and show the strategies you have used.

[(654 ÷ 109) × 43 × (652 ÷ 326)] ÷ 12 − (34 × 5 − 127) =

Which of the following is a better estimate? Circle the best choice in the right-hand column.

(a) For 72 × 28? 72 × 30 or 70 × 28

(b) For 16 × 1, 897? 20 × 1, 800 or 15 × 2, 000

(c) For 6 × 1, 029, 714? 6 × 1, 000, 000 or 10 × 1, 029, 714

two expressions are equal.

Complete the following definition. An equation is a statement that__________________.

5x+10y

Write the expression for: The value in cents of x nickels and y dimes.

Make up a short word problem that builds the given expression as an answer in the given context. Be sure to make clear what the x represents: 180+x 60 as the time needed to complete a trip between three cities.

Give an Teacher’s Solution with Algebra to the following problem. The lengths of the sides of a quadrilateral are four consecutive whole numbers. If its perimeter is 86 inches, find the length of the shortest side.

a.the position

b.quotient

c. distributive

d. expressions

Fill in the missing word.

(a) “Place value” refers to the fact that, in decimal numbers, the value of a digit depends on its .

(b) In the equation 15 ÷ 3 = 5, the number 5 is called the .

(c) The Mental Math calculation 3 ×21 = (3×20) + (3×1) uses the Property.

(d) An equation is a statement that two are equal.

=14²-1=195

Use the identity (a + 1) · (a − 1) = a 2 − 1 to calculate 13 × 15. Show all the steps clearly.

=14²-4=192

Use the identity (a + 2) · (a − 2) = a 2 − 4 to calculate 12 × 16. Show all the steps clearly.

(30-1)62 =30²-2×30×1+1²

=900-60+1=841

Use the identity (a − b) 2 = a 2 − 2ab + b 2 to calculate 292 . Show all the steps clearly.

Give a picture proof to the identity (a+b) 2 = a 2 + 2ab+b 2 by illustrating it on a rectangular array with labels indicating both sides of the equality are visible.

Is 257 prime or composite? By the primality test, what is the largest prime you have to check?

distributive

A student claims that 17x + 12x = 29x because of “adding like terms”. What arithmetic property is the student using?

a. commutative

b.associative

c.distributive

State which arithmetic property is being used:

(a) 3y + b + c = b + c + 3y Property:

(b) t × (v × q) = (t × v) × q Property:

(c) 2 × (g + h + i) = 2g + 2h + 2i Property:

Use mental math to compute the following. Write down your answer in a way that clearly shows the steps involved in solving the problem mentally.

51²

Which is larger: 8^5 or 2^14? Justify your answer using comparisons of expressions that contain identical bases or identical exponents.

Simplify as much as possible. Write out every step neatly – this will reduce errors.

Use the exponent rules to simplify as much as possible. State the rules or properties used at each step.

Calculate the following using mental math. Include intermediate steps to show your thinking.

(a) 16 × 14

(b) 16 × 32

(c) 49²

Give a full Teacher’s Solution for the following word problem. Draw a bar diagram and also show the algebra.

Harry bought 155 oranges for $35. He found that 15 of them were rotten and threw these away. He sold all the remaining oranges at 7 for $2. How much money did he make?

Give a complete Teacher’s Solution using algebra for the following: Ann has twice as many stickers as Ben. How many stickers must she give to Ben so that they each have 15?

2x

2x+1

Fill in the blanks:

An even number is ____________ and an odd number is________.

Give a picture proof (complete with labels) for the fact that “the sum of two even numbers is an even number”.

Give an algebraic proof for the fact that “the sum of two even numbers is an even number”. List all the steps and the properties used.

Give an algebraic proof for the fact that “an even number times any whole number is even”. List all the steps and the properties used.

Circle the numbers below that divide the number 117, 645.

3 4 5 8 9 11

Find all digits a so that the number 5a3141132 is divisible by 6, but not divisible by 9. Explain your reasoning.

Put one digit in the blank to make an 8-digit number that is divisible by 11: 58, 3 2, 492.

Find the prime factorization of 3564.

14×10=2²x5×7

Use mental math to work the following. Factor 140 =

Find 101! ÷ 100!. (Hint: You are not expected to multiply 100 numbers!)

Determine how many zeros are at the end of the decimal form of 55!.

2,3,5,7,11,13,17,19,23,29

List the first 10 prime numbers.

What is the largest prime less than 236? (Show your reasoning.)

Use the prime factorizations to find GCF and LCM of 28 and 63.

homework problem

• Chocolate milk is served every 2 days.

• Corn is served every 4 days.

• Pizza is served every 7 days.

The LCM of 2, 4, and 7 is the smallest number that is a multiple of each of these numbers.

1. Prime factorization of each number:

   • 2 = 2

   • 4 = 2²

   • 7 = 7(7 is already prime)

2. LCM is found by taking the highest power of each prime factor:

   • The highest power of 2 is 2².

   • The highest power of 7 is 7.

Thus, the LCM is:

LCM=2²×7=4×7=28

The smallest number of days until all three are served again is 28 days.

At a summer camp, chocolate milk is served every other day, corn is served every 4 days and pizza every 7 days. Today all three were served. What is the smallest number of days until all three are served again?

Use the Euclidean algorithm to find the greatest common divisor of 210 and 560. Show all your work.

Use the Euclidean algorithm to find the greatest common divisor of 210 and 560. Show all your work