5 Written Questions
5 Matching Questions
- Primes 1
- Remainders 1
- Factor Pairs
- Simplifying Exponential Expressions 1
- Relationships in Even Sets, Consecutive Multiples and Integers
- a - All sets of consecutive integers are sets of consecutive multiples
- All sets of consecutive multiples are evenly spaced sets
- All evenly spaced sets are fully defined if these three parameters are known:
1. The smallest (first) or largest (last) number in the set
2. The increment (always 1 for consecutive integers
3. The number of items in the set
- b On simple problems, pick numbers.
- add the desired remainder to a multiple of the divisor
- ex: need a number that leaves a remainder of 4 after dividing by 7; (7*2) + 4 = 18
- c - pairs of factors that yield an integer when multiplied together
- how to find a factor pair of X:
1. Make a table with 2 columns labeled "small" and "large"
2. Start with 1 in the small column and X in the large
3. Test the next possible factor of X
4. Repeat until the numbers in the small and large columns converge
- d Any positive integer larger than 1 with exactly two factors: 1 and itself.
- 1 is not considered a prime
- the first prime is 2, only even prime
- first ten primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
- e 1. Only if expressions are linked by multiplication or division. CANNOT simplify expressions linked by addition or subtraction.
2. Only if linked by multiplication or division if they have a base or exponent in common.
3. Use exponent rules (book 1, pg. 68).
4. Factor whenever bases are the same.
5. Factor when the exponent is the same and the terms have something in common.
6. Study chart in book 1, pg. 70.
5 Multiple Choice Questions
- 1. If you add a multiple of N to a non-multiple of N, the result is a non-multiple of N (same for subtraction).
2. If you add two non-multiples of N, the result could either be a multiple or non-multiple of N.
- Whole numbers that are either positive or negative, including zero.
- adding, subtracting and multiplying result in integers.
- result of division is called a quotient.
- "divisible" = no remainders
- "divisor" or "factor" is a number that can evenly be divided into another
- NO GURANTEES!
- can result in odds, evens or non-integers
- odd divided by any number will never be even
- odd divided by even will never equal an integer
- 1. For any set of consecutive integers with an ODD number of items, the sum of all integers is ALWAYS a multiple of the number of items.
- ex: 1+2+3+4+5=15, multiple of 5
2. for any set of consecutive integers with an EVEN number of items, the sum of all items is NEVER a multiple of the number of items.
- ex: 1+2+3+4=10, not a multiple of 4
3. Use prime boxes to keep track of factors of consecutive integers.
- Factors divide into an integer "X" and are therefore less than or equal to X. Positive multiples multiply out from X and are therefore greater than or equal to X.
- all integers have a limited number of factors
- all integers have an infinite number of multiples
5 True/False Questions
Perfect Squares → 1. Adding: when multiplying two numbers with the same base, combine exponents by adding
2. Subtracting: when dividing two numbers with the same base, combine exponents by subtracting them
3. Multiplying: when raising a power to a power, combine exponents by multiplying
4. Dividing: a negative exponent means putting the term with the exponent as the bottom of a fraction and making the exponent positive
- when there is a negative exponent, think reciprocal
5. Exponent of 1: any number that doesn't have an exponent implicitly has an exponent of 1
6. Exponent of 0: any non-zero base raised to the power of 0 is equal to 1
7. Fractional exponents: the numerator tells us the POWER to raise the base to, and the denominator tells us which ROOT to take
Greatest Common Factor (GCF) & Least Common Multiple (LCM) → GCF: the largest divisor of 2+ integers
LCM: the smallest multiple of 2+ integers
- use a venn diagram to find GCF and LCM
1. Factor both numbers into primes
2. Place common factors in the shared area (incl. copies)
3. Place non-common factors in the sides
4. GCF = product of primes in the shared area
5. LCM = product of all primes in the diagram
- if no primes are in common, the GCF is 1 and the LCM is the product of the two numbers
When to use Prime Factorization → 1. Determine if x is divisible by y
2. Determine the greatest common factor of two numbers
3. Reducing fractions
4. Finding the least common multiple of a set of numbers
5. simplifying square roots
6. Determine the exponent of one side of an equation with integer constraints
Exponent Rules → 1. All perfect squares have an odd number of total factors
- any integer that has an odd number of factors MUST be a perfect square.
2. The prime factorization of a perfect square contains only even powers of primes.
- any number whose prime factorization contains only even powers of primes must be a perfect square.
3. Same rules apply to perfect cubes, quads,etc., and factorization needs to have multiples of that "perfect N".
Divisibility & Addition/Subtraction → All of the following say the same thing:
- X is divisible by Y = Y is a divisor of X
- X is a multiple of Y = Y divides X
- X/Y is an integer = X/Y yields a remainder of 0
- X=3(n), n being an integer = Y "goes into" X evenly