Review: Integers

Integer Properties - Strategies for Questions (review / summary)

1. *Make sure you're dealing with integers!* Test FREQUENTLY sets traps in problems with variables for those who automatically assume that the variables represent integers. Thus, do NOT automatically apply any integer properties if not guaranteed that the variables of the numbers of the problem are in fact integers. If the question identifies the variables as integers explicitly, or implicitly through specific words like even, odd, prime etc. (note: *prime numbers are always positive integers*) 2.*Unless given clues like "prime" or "remainders", always consider zeros and negatives!* Another common trap - Never assume a number is positive or even and always consider ALL applicable categories of numbers - negatives, zero, odd (and if not dealing with integers, don't forget fractions and decimals). 3. *Remember the many interchangeable ways to talk about factors*. 13 is a factor 78 13 is a divisor of 78 78 is divisible by 13 78 is a multiple of 13 13 is part of the prime factorization of 78 ==> All represent the same fundamental mathematical fact, and the test uses these ideas interchangeably, and expects you to use them in interchangeably as well. 4. *One is NOT a prime number and 2 is the only even prime number* - these facts are very frequently tested! Memorizing all the primes below 60 will save a lot of time. 2,3,5,7, 11,13,17,19, 23,29, 31,37, 41,43,47, 53,59 5.*Prime factorization is the single most important strategy for unlocking problems dealing with integer properties*. Prime factorizations unlock all kinds of secrets about numbers. - if a number >100 appears in any question, especially related to factors, multiples or division, finding the prime factorization is likely the key to answering the question 6. *Any number squared is positive and 0² = 0*. To save time, memorize perfect squares of the first 10 integers and helpful to memorize the perfect squares of 11 through 15 as well. 1,4,9,16,25,36,49,64,81,100 121,144,169,196,225,256,289,324,361,400 7. *Ensure you can find LCM and GCF* - and ensure you are NOT falling into trap for using the LCM formula 8. *Use logic and PLUGGING IN for Even / Odd questions as needed.* When plugging in use simple numbers (if two variables, plug in 1 for odd and 2 for even) and remember to test all four cases: a) both even, b) both odd, c) x = even, y = odd and d) x = odd , y = even 9. *Consecutive number questions often appear in variable forms* - recognize the algebraic expressions but ensure the *expressions relate to integers* (b/c if not integers, doesn't apply!). When dealing with complex equations, factor / simplify and look for a patter representing consecutive numbers (see example in figure - factoring out t and rearranging factors gives us the product of three consecutive numbers - ONLY if we know that t is an integer!) 10. *For all questions involving a remainder, two powerful strategies are: (a) listing possible dividends and (b) rebuilding the dividend formula D = S•Q + r*.

Perfect Squares - Example

*The exponents of the prime factors of a square must all be even ==> given an unknown number in its prime factorization form where all exponents are even, number *must be a perfect square*

Divisibility when quotient is not an integer

There are two separate, but perfectly correct, mathematical procedures. Need to understand how to employ either option, but you never will have to decide which one to use - question will always make clear which one b/c will refer to result of division with remainders or fractions / decimals (Review "Arithmetic - Decimals and Fractions" set for conversions, and cards in this set for more detail on division with remainders)

Prime Factorization

*Finding the prime factorization is the most important skill when dealing with integers* - critical to understand and practice! Prime factorization examples: 9 = 3•3 10 = 2•5 12 = 4•3 = 2•2•3 15 = 5•3 24 = 6•4 = 3•2•2•2 = 3•2³ 100= 10•10 = 2²•5²

Perfect Squares

*The exponents of the prime factors of a square must all be even*. This is because each factor of an integer appears twice in its square ("doubled up" since the number appears twice when squared) N² = N•N = (factors of N) • (factors of N) e.g. prime factorization of 12² = 144 = 12•12 = [2•2•3]•[2•2•3] (i.e., each factor of 12, appearing twice because there's 2 12s) = 2⁴• 3² *Given an unknown number in its prime factorization form where all exponents are even, number must be a perfect square*.

Uses of Prime Factorization

1. Identify number of factors of any given number, including subsets (number of +ve factors, odd / even factors etc.) 2. Identify perfect squares based on all even exponents and/or odd number of factors 3. Find GCF ==> Find LCM

Even and Odd Integer Properties *(frequently tested!)*

1. Remember: zero is neither positive, it's not negative, but it IS an *even integer*. 2. So, if told that x is even or y is odd, don't automatically it's positive! 3. So if told x is even or y is odd, you absolutely know those numbers are integers! 4. Very important idea to recognize that prime factorization of an even number *always* contains at least one power of 2 as one of the prime factors! And prime factorization of *any odd number* is guaranteed there to *never* include a factor of 2. 5. Algebraic expressions of *2k + 1 and 2k-1 are odd for every even integer k* - important to recognize.

Consecutive Integers - Basic Facts (frequently tested)

1. VERY important to know this! (e.g., 4 consecutive integers - 1 will always be divisible by 4) 2. Be careful - sum divisibility ONLY works for odd numbers 3. Makes sense when you of number order - in a set of 4 consecutive integers, one of the even numbers would have to be divisible by 4 and at least one number is divisible by 3 (could be more but this depends on which numbers - but we know that least one is divisible by 3)

Even & Odd Numbers - Example 1 using PLUGGING IN to replace logic!

Logic is usually quicker (fig 3) but if stumped, can approach a question by plugging in (fig 2)

Least Common Multiple (LCM)

AKA Least Common Denominator Smallest multiple shared by two numbers. The product of two numbers is always ONE of the common multiples, but often not the LEAST common multiple for those two numbers. For small numbers, can list all multiples out, identify the common multiples and then take the lowest of those numbers. For larger numbers, 1. Find GCF (prime factorization ==> factors with highest powers in common ==> Multiply 2. Write each number in the form of GCF•x 3. LCM = GCF•(x of one #)•(x of second #) Symbolic representation of LCM of M and N = GCF•[N/GCF]•[M/GCF] NOTICE - this simplifies to *GCF•[N]•[M/GCF]* OR *GCF•[N/GCF]•[M]* (see GCM-LCM formula for more)

Greatest Common Factor (GCF)

Aka Greatest Common Divisor (this is how to find GCF - application of this is to find LCM ==> see LCM card for details) Find GCF of smaller numbers (ex. in top fig): 1. List all factors of both numbers 2. Identify the common factors in both lists 3. Find the largest # in the common factors For large numbers (ex. in bottom fig): 1. Find prime factorizations of each 2. Find highest powers in common (i.e. overlap of exponents for each factor X = highest power of each factor that BOTH have) 3. Multiply the factors using the highest powers in common (in example, that's 3 factors of 2, 0 factors of 3 and 1 factor of 5)

Prime #s: Testing if a number >100 is prime

An odd number less than 100 *must be prime if it is not divisible by one of the prime numbers less than 10* (i.e., 2,3,5,7). If divisible by 1 or more, not a prime number e.g., For all positive integers N such that 80≤N≤90, how many prime numbers are there? 1. Excluding even numbers (divisible by 2) and 85 (divisible by 5) ==> 81, 83, 87, 89 ==> exclude 81 (b/c = 9•9) 2. Test divisibility by 3 on rest ==> exclude 87 (8+7=15, so divisible by 3) 3. Need to test divisibility by 7 ==> starting at 7•10, multiples of 7 are 70, 77 84, 91 ==> so neither 83 nor 89 are divisible by 7 and are thus the only 2 prime numbers in this example

Odd & Even Integers - Trick Questions

Be careful - might be tempted to go with C but we were not told that Q / R were integers!! Since can get the answer when they are fractions, the answer is E because all answers assume they are integers *NEVER assume numbers are integers unless explicitly told as much!* ==> MAJOR strategy for any number property question

Multiples - Practice Q

Bunch of multiples of P ==> so P is a factor of all of these numbers. And we want to know which of the answer choices may equal P. Note that last info provide (15•K is a multiple of P) is redundant because any multiple of K is going to be a multiple of P. But the sums are interesting: K, K + 200 and K + 350 are all multiples of P ==> differences among these will also be multiples of P ==> so 200 and 350 are also multiples of K (b/c [(K+200) - K] = 200) And since 350 & 200 are also multiples, I could subtract those to get another multiple of P: 350-200 = 150 Another subtraction: 200-150 = 50 ==> another multiple of P. Thus, P is a factor of all four of these numbers: 50, 150, 200, and 350. Looking at the answer choices: - eliminate anything bigger than 50 (b/c anything > 50 can't be a factor of 50) - eliminate 20 b/c except for 200, none of the factors are divisible by 20 Answer is either 25 (B) or 75 (C) but only 25 divides 50, 150, 200, and 350 evenly ==> so it's the only one that is clearly a factor of all four (50, 150, 200 and 350), and is thus the correct answer

Even & Odd Numbers - Example 2 using PLUGGING IN to CONFIRM logic

Logical reasoning: the only way that the sum will be odd is if Q is an odd number since 4P MUST be even. So, Q must be odd but we can't draw a conclusion about P because we know 4P is even but can't discern what P is. In this case, testing cases might be helpful to confirm logic

Common Divisibility Rules (2,3,4,5,6,9)

Divisibility rules are *very commonly tested*, but especially focus on rule for *3* and ** 2 & 5 ==> only need to look at last digit, i.e. ones place (even for 2, 0 or 5 for 5) 4 ==> need to look at last two digits, i.e., tens and ones place (even? then work through divisible by 4) 3==> need to see if a different number, i.e. the SUM of all the digits in the number, is divisible by 3 (e.g., 135 => 1+3+5 = 9 => divisible. BUT 734 => 7+3+4=14 => not divisible) 9==> similar to rule for 3, check whether SUM of the digits is divisible by 9 (e.g., 1296 ==> 1+2+9+6 = 18 ==> divisible by 9. BUT 6==> any even number that is also divisible by 3, i.e., need to check divisibility rules for 2 & 3 Note: testing divisibility by 7 or 8 is uncommon, which is why not covered Example (for all rules): 3072: 2: YES 3: YES (3+7+2 = 12) 4: YES 5: no 6: YES 9: NO (12 is divisible by 3 but not 9) Note: because it's divisible by 3 AND 4, it's also divisible by 12! (3•4)

GCM-LCM Formula (IGNORE IF CONFUSING)

For last figure - i.e. divide P OR Q by denominator FIRST!! Recall: LCM of M and N = GCF•[N/GCF]•[M/GCF] NOTICE - this simplifies to *GCF•[N]•[M/GCF]* OR *GCF•[N/GCF]•[M]* (see GCM-LCM formula for more)

Prime Factorization - Finding positive factors of large numbers >100

For smaller numbers, numbers <100, simply list the factor pairs For larger numbers, use the following four steps: 1. Find prime factorization of the number with exponents 2. Make a list of ALL exponents (don't forget to use exponent of 1 for #s "without" an exponent AND include all duplicates!) 3. Add 1 to EACH exponent to get a new list 4. Multiply numbers in list from step 3 Note: *if excluding 1 and the number itself, subtract 2 from this number to get # of positive factors* Note 2: If only want ODD factors, ignore all factors of 2. There is no direct way to get even factors, but can calculate (all factors - odd factors = even factors) e.g., How many factors does 8400 have? 1. Prime factorization: 8400 = 84•100 = 7•12•10•10 = 7•2⁴•3•5² 2. Exponents: 1, 4, 1, 2 3. Add 1: 2, 5, 2, 3 4. Multiply: 2•5•2•3 = 60 ==> 60 different factors, including 1 and 8400 ==> 58 factors without 1,8400 ==> 24 odd factors (exponents are 1,3,2 ==> add 1 to each to get 2,4,3 ==> multiply to get 24) ==> 36 even factors (All-odd = 60 - 24)

Remainders - relationship to divisor

Formula provides connection between the integer quotient (Q) and the mixed-numeral quotient (Q + r/s) - this means, that if a quotient written in mixed-numeral or decimal form ==> *non-integer part of the quotient is remainder over divisor (r/s)* - subtle but important point e.g., if D/S = 21876.375 ==> Q = 21876 and r/S = 0.375 = 3/8 (NOTE: this does NOT mean r=3 and S = 8 as they could be ANY fraction equivalent to this, i.e. 6/16, 9/24, etc. But if we knew r OR S, could find the other since we know the ratio) Key skill for remainders: generating possible dividends that, when divided by a certain divisor, yield a specific remainder (see next card for example)

Importance of LCM + Relation to factors

May be asked directly, may be needed when adding or subtracting fractions (LCM = LCD) or may be asked indirectly, if asked about matching things in sets of different sizes (e.g., hotdogs are sold 12 to a pack but hot dog buns are sold 8 to a pack. What is the least number of buns that you'd have to buy so that you'd have a bun for every hot dog?)

Dividing Even and Odds

No general rules for evens and odds with division, largely because division of two integers may not even equal another integer but be a fraction or a decimal When the quotient of two integers does happen to be another integer, *whether that quotient integer is even or odd depends on what factors cancel and what factors remain* but it cannot be predicted simply by whether the two numbers divided were even or odd. Can't make predictions but there are some general patterns: i) E÷E = E or O or non-integer ii)O÷O = O or non-integer iii)E÷E = E or non-integer iv) O÷E = non-integer (ALWAYS!)

Factors and Divisors / Divisibility and Multiples

Note: assume A, B and C are positive integers: Important to keep in mind that 1 is a factor of every positive integer! If C/A = B ==> then A is a *divisor* of C, because it divides evenly into C and the output, B, is an integer. If A is a divisor of C, then C is *divisible* by A Note: *factor and divisor mean the same thing* - factor is in context of multiplication while divisor is in context of division but essentially they mean the same thing. *Divisible by and factor of* are the same thing - again, different contexts. Also important to know: *multiple is the inverse relationship of factor* (if a is a factor of b, then b is a multiple of a) Given positive integers P, Q: If P is a multiple of Q ==> P is the product of Q multiplied by some positive integer. In other words, *Q is a factor of P and a divisor of P* (and since multiple is the inverse relationship to factor ==> *if Q is a factor of P, then P is a multiple of Q*) e.g., 63 and 888 are multiples of 3 while 70 and 1255 are multiples of 5 Note on zero: 0 is a multiple of every integer (because any integer multiplied by zero is zero). But 0 is neither positive nor negative, so if looking for the first positive multiple of 5, you would start from 5, not 0 because 5*0 means I have no fives at all and it isn't positive. Thus, the terms are interchangeable b/c the same mathematical info is provided by all: 1. 8 is a factor of 24 2. 8 is a divisor of 24 3. 24 is divisible by 8 4. 24 is a multiple of 8

Consecutive Integers - Algebraic Representation (+ example)

Notice that the example Q gives sth that is deceptively confusing but factoring and re-writing makes it clear that this is a product of four consecutive integers And then can answer the question based on facts about a product of 4 consecutive integers - 2 are even, 2 are odd, one is divisible by 4, at least one (if not more) is divisible by 3 I. For a number to be divisible by 8 ==> need a factor of 2 and a factor of 4 ==> YES II. For a number to be divisible by 12 ==> need a factor of 3 and a factor of 4 ==> YES 111. For a number to be divisible by 18 ==> need a factor of 2 and TWO factors of 3 ==> MAYBE! (not necessarily - we only know we have AT LEAST one factor of 3)

Prime Numbers

Prime numbers act a "building blocks" of all other positive integers and are in some sense, the most important positive integers. (building blocks b/c of prime factorization) There are 8 prime numbers less than 20: 2,3,5,7,11,13,17,19 IMPORTANT: *1 is NOT a prime #* ( b/c any prime number must have TWO factors (1 and itself) but 1 has only ONE factor (itself)) and *2 is the only EVEN prime number*

Perfect Squares - Counting Factors

Remember: all the powers of the prime factors of a perfect square are even. So all numbers in the list of exponents will be even ==> when we add 1 to each number, this list will now be all odd numbers ==> product will always be odd Thus, *a perfect square always has an odd number of factors* Can also think about this in terms of pairs (see fig.) - last "pair" has only one number, so there's an odd number of factors ==> only happens in perfect squares because otherwise, there will be a pair!

Multiples - Details

Reminder: factor, divisor, multiple and divisible are interchangeable b/c give same mathematical info in different contexts. Example: 7 is a factor of 91 7 is a divisor of 91 91 is divisible by 7 91 is a multiple of 7 Ideas to keep in mind for multiples: 1. 1 is a factor of every positive integer AND every positive integer is a multiple of 1 (note: this means that multiples of 1 are ALL the positive integers, which is why we don't often refer to this) 2. Every positive integer is both a factor of itself, AND a multiple of itself. (i.e., given any number, the largest factor AND the smallest multiple is the number itself) 3. Can get the first P multiples of a number r by multiplying r by [1,2,3,.....P]. (i.e., *add r to itself successively P times to get P multiples*) E.g., need first five multiples of 12 ==> multiply [1,2,3,4,5] to get: 12, 24, 36, 48, 60 (i.e., add 12 to itself successively 5 times) ==> see figure #3 for generalization of adding/subtracting r to get more multiples of P (essentially, *if P is a multiple or r, then any multiple of P is a multiple of r* 4. Given 2 or more multiples of r, *can add and subtract ANY two multiples to get more multiples of r* (see figure #4) 5. If *P* and *Q* are both multiples of *r*, then their *product P•Q must also be a multiple of r* IN SUMMARY: Can ADD, SUBTRACT or MULTIPLY to get more multiples of r (but can NOT divide!) e.g., 24 and 80 are both multiples of 8 Add: 24+80 = 104 ==> also a multiple of 8 Subtract: 80 - 24 = 56 ==> also a multiple of 8 Multiply:24*80 = 1920 ==> also a multiple of 8

Remainders - unexpected application of remainders and dividers

SHOWS UP FREQUENTLY! 1.*Generating possible dividends that, when divided by a certain divisor, yield a specific remainder* (see fig - simplest answer is adding 5 to 12, and can keep adding / subtracting 12 to get more). i.e., Similar to how we can get from one multiple to the next by adding the number, we can also get from one of these numbers to the next by adding the number. Thus, if told that 1997 is a number that, when you divide by 12 has a remainder of 5 ==> could find other numbers of this set by adding or subtracting 12. 2. *Recognize that if the divisor is larger than the dividend, integer quotient is zero and the remainder equals the dividend.* (tested VERY frequently!). e.g., What is the smallest positive integer that when divided by 12, has a remainder of 5? Answer is NOT 17 (common mistake), but is actually 5 ==> Q is 0, R=5. 3. *Rebuilding the dividend* - D = S•Q + r (IMMENSELY IMPORTANT)

Consecutive MULTIPLES

Special cases of evenly spaced sets with a pattern where *nth number in the set = n + multiple•(n-1)* In this example, it's 135 + 5*(40) = 135 + 200 ==> different between first and last number is thus 200.

Finding Positive Factors

The easiest way to find *every factor* of a number is to list the *factor pairs* - i.e., the pairs of numbers that have a product of x *Don't forget the factor pair of 1* (a factor of every number) *and x* (every number is a factor of itself when multiplied by 1) From 1, go through 2,3,4,5 etc. (skipping numbers that aren't factors). Stop when you get to highest factor of x•x OR when you reconnect with a previous number (e.g., for 20, 4•5 => reconnect with it when you get to 5, b/c it's equal to 5•4 ==> factorization stops at 4) e.g., Find all positive factors of 36 - factor pairs are every pair of number where product = 36: 1,36 2,18 3,12 4,9 (skip 5) 6,6 (end ==> since this is x•x) ==> 9 positive factors (includes 1, itself, but don't count 6 twice) For small numbers, numbers less than 100, we can count the positive factors simply by making a list of factor pairs. For larger numbers (e.g., 12,600), see alternative method *Negative factors* are just repeats of the positive factors, which is why almost never tested on this (will show up ONLY on advanced questions) - so 36 has 9 positive factors and also has 9 negative factors for a total of 18 factors. Look out for this on advanced Qs when "positive" is not explicitly mentioned with factors!

Prime Factorization & Multiples/Factors

The prime factorization of a number is like the DNA of the number, revealing all its essential ingredients. *Any factor of Q must be composed only of prime factors found in Q.* Recall synonymous statements of factors, divisors, divisible and multiple ==> can now add one more as follows: 1. r is a factor of Q 2. r is a divisor of Q 3. Q is divisible by r 4. Q is a multiple of r 5. *Every prime factor of r is included in the prime factorization of Q* In the example, the distinction between the two (25 and 85) that are NOT factors is important to distinguish - 25 is a difference of instances (has TWO instances of 5 vs. 1) while 85 has a new factor that isn't in the original number (17) ==> both of these cases means a # can NOT be a factor

Prime #s: Important Prime Facts

VERY IMPORTANT - more than 50% of the time when test is asking questions about prime numbers, at least one of these two facts is hidden somewhere in the question!

Integer Properties - Intro

VERY commonly tested so important to understand integer properties. Reminder: integers are the non-fraction numbers, which includes all the positive and negative whole numbers, as well as 0 For integers, need to understand key mathematical terms of *factor*, *divisor*, *multiple* and *divisible* which are essentially interchangeable ways to say the same thing - key to know b/c test will use all (see individual cards for more info on each)

Remainders

When we divide any integer by one of its divisors, the quotient is always an integer and that integer is always another one of the number's factors. e.g. 21 ÷ 3 = 7 When dividing one number by another number that is not one of the the factors, result is a fraction or a mixed numeral. e.g., 17÷5 = 3+2/5 ("mixed numeral quotient") Alternatively, if we want to keep everything in terms of integers (e.g., dealing with groupings or sets and need to know how many full sets and how much left over), we would use *integers for quotient and remainder*. i.e. 5 goes into 17 THREE times with a remainder of 2 (3 is the "integer quotient"). Note on terminology: in this case, "divisor" here ≠ factor ==> 6 is not factor of 20! With remainders, it's just literal

Multiplying Even and Odds

like = same as factors Unlike = even The reason for this is b/c product includes ALL combined factors. Thus, E•O = E b/c product includes ALL factors of both and thus has a factor of 2 from the E ==> product is E IMPORTANT implications of this! (see next card)