Number System

There are many odd prime numbers but there is only one even number that is prime, and that number is ____.

2

Example Question: To find the average of (x + 3), x - 6 and 4x - 3: Solution: Add the group of numbers: (x + 3) + (x - 6) + (4x-3) = 6x - 6 Then divide by __: 6x -6 / 3 = 2x - 2

3

The set {7, 8, 9, 10} represents a set of ______.

4 consecutive integers

The set {12, 14, 16, 18, 20} represents a set of _____.

5 consecutive integers

Some problems might give you an average and ask you to find the missing score. Example: If John's scores are 70, 84, 74, and 76, what must he get on his next test to average 80 for the five tests. Solution: Total sum required: average desired (80) x number of tests (5). Total points already accumulated = 70 + 84 + 74 + 76 = 304. Total required less total accumulated: 400 - 304 = ___ must be the score on next test.

96

A problem to test your understanding of the effect that exponents have on fractions on the CLEP exam might be as follows: Example: If P < 0, what is true of p³? (A) p³ > p (B) p³ < 3 (C) p³ = p (D) p³ = p² + p (E) cannot be determined from given information Solution: Notice that it does not say that P is a whole number, only that P is negative. If P were a whole number, then raising a negative whole number to the third power would make a smaller or equal negative number. If P were a negative fraction greater than - 1, then raising it to the third power would make a larger negative number. If P were equal to - 1, then p³ would also be - 1. We could eliminate choice D immediately, since it violates the laws of exponents. Therefore, non of these choices is correct for all cases, and the correct answer is ____.

E

Example Question: If b is an odd number, which of the following is (are) also odd? I. 2b + 1 II. 3(b + 1) III. 3b + 1 Solution: Remember our rules: -In a sum, the only way to get an odd number is if one number is odd and the other number is even. -In a product, the only way to get an odd number is if both numbers are odd. Therefore ____ is odd.

I (you can also use substitution to solve this problem)

Example Question: Which of the following must be odd? I. The product of two odd numbers II. The sum of an odd and an even number III. The product of two even numbers Solution: Remember our rules -In a sum, the only way to get an odd number is if one number is odd and the other number is even. -In a product, the only way to get an odd number is if both numbers are odd. Therefore ____ and ____ must be odd.

I, II (you can also use substitution to solve this problem)

______ are whole numbers: 1, 7, 23, 48, and 526 are examples.

Integers

There are times in mathematics when we want to ensure that our answer be positive. If a number is negative, we'll take its positive counterpart. This is called ____, and is represented by the symbol "I I".

absolute value

Integers that are divisible by 2 are ____ integers. Those not divisible by 2 are ____, except for 0.

even, odd

-In a sum, the only way to get an odd number is if one number is ____ and the other number is _____. -In a product, the only way to get an odd number is if both numbers are ____.

even, odd, odd

Example Question: If X represents and even number and Y represents an odd number, which of the following are not odd? I. X + Y + 1 II. 2Y + 3 III. XY + Y Solution: I is ____ because you have the sum of an odd and an even which is odd. Adding one more makes its even. II is ____ because the product of 2 and Y must be even, and the sum of an even and an odd is odd. III is ____ because the product of an odd and an even is odd. If you forget the rules, just substitute and even number for X (2) and and odd number for Y (1) and evaluate each expression.

even, odd, odd (-In a sum, the only way to get an odd number is if one number is odd and the other number is even. -In a product, the only way to get an odd number is if both numbers are odd.)

The rules for combining singed numbers are: -If the signs are the same, add the numbers together and take the ____ sign. -If the sings are different, take the difference of the 2 numbers and the sign of the ____.

larger

Example Question: Find the product of 3 consecutive integers whose sum is 30. Solution: First, we need to divide 30 by 3. We get 10. The numbers will be 10, 1 _____ than 10, which is 9, and 1 ____ than 10, which is 11. Their product is 9 x 10 x 11 = 990.

less, greater

To find the arithmetic mean, or just the ____, of a group of numbers, you add the numbers together and divide by the number of numbers in the group. Example: A 10 question quiz resulted in the following scores. Find the mean test score from the following set of numbers: {3, 4, 5, 5, 6, 7, 7, 7, 10} Solution: There are 9 numbers whose sum id 54. The mean is equal to 54/9 = 6.

mean

The _____ defines a middle point, such that there are an equal number of numbers in our set below this point as there are above this point. Example Set: {3, 4, 5, 5, 6, 7, 7, 7, 10} Solution: The median score would be 6.

median

The ____ is the most common occurrence in a set of numbers. Example Set: {3, 4, 5, 5, 6, 7, 7, 7, 10} Solution: The most common number is 7. 7 is the mode.

mode

Integers can be both positive and negative. Integers include all the numbers that we know as Whole Numbers, but in addition, they also include the opposites of these numbers, the _____ numbers.

negative

Example Question: If a = b², what happens when b is tripled? Solution: First we substitute 3b for b. Remember to square 3b. Therefore, a is ____ times larger when b is tippled: a = (3b)², a = 9b².

nine

Some problems on the CLEP exam might ask you to deal with an algebraic expression and decide if its is odd or even. Example: If a and b are positive odd numbers, is the expression a² + 3ab + b² odd or even? Solution: Since a is odd, a² is odd. Since b is odd, b² is odd. Since a and b are odd, 3ab would be 3 times an odd, or odd. Therefore, we would a sum of 3 odd numbers which would be ____. Note: If the expression were a² + 2ab + b² we could recognize it as the square of: (a + b)² = a² + 2ab + b². If a and b are odd, then a + b is even and an even number squared is even.

odd

-An odd number with any exponent produces an ____ number. -An even number with any exponent produces an ____ number.

odd, even

The rules for multiplying signed numbers are: + x + = + + x - = - - x + = - - x - = + If the signs are the same, the product is _____; if the signs are different, the product is _____.

positive, negative (the rules for dividing signed numbers are the same as the rules for multiplying; any division problem can be written as a multiplication problem)

The numbers 5, 2, and 2 are the _____ of 20. They are prime numbers that will divide evenly into 20.

prime factors

A number is a _____ if it can only be divided evenly by itself and 1. Written as a product, a prime number can only be written as a product of itself and the number 1.

prime number

On a number line, numbers get larger as you move to the _____, smaller as you move to the _____. Example: - 2 > -6.

right, left

The listing of 1, 7, 23, 48, and 526 (for example) represents a ____ of integers. It is not the set, or listing of all integers, but its is a set of integers and is represented as: {1, 7, 23, 48, 526}.

set

Example Set: {1, 7, 23, 48, 526} 48 and 526 represent a _____ of this set (they are even numbers) and this subset is represented as {48, 526}.

subset

In some average problems, the numbers may represent different quantities of averages themselves. In this case we have a _____ average. Example: If 8 shirts of one kind cost $10.00 each and 4 of another kind cost $16.00, what is the average cost of all shirts? Solution: Here we must multiply to find the total sum first. 8 x 10 = $80.00 4 x 16 = $64.00 Total sum is $80 + $64 = $144; 12 shirts cost $144. Therefore, the average for all 12 is $12.

weighted