GMAT (Fach) / Quantitative - Arithmetic (Lektion)

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• Integers Whole, no fractional part, can be + and - SUM of 2 integers = ALWAYS INTEGER DIFFERENCE of 2 integers = ALWAYS INTEGER PRODUCT of 2 integers = ALWAYS INTEGER BUT QUOTIENT of 2 integers = SOMETIMES INTEGER Only integer if remainder = 0
• Rules of Divisability: A number is divisible by 2, if ..the integer is even 1234567 : 2 = ODD, non-divisible 1234568 : 2 = EVEN, divisible
• Rules of Divisability: A number is divisible by 3, if the SUM of the integer's digits is divisible by 3 72 -> 7+2 = 9 IS DIVISIBLE BY 3 73 -> 7+3 = 10 is not divisible by 3
• Rules of Divisability: A number is divisible by 4, if the Integer is divisable by 2 TWICE 28 -> 28:2=14, 14:2=7 -> because you can divide it by 2 TWICE for a large number, if the LAST TWO Digits are divisible by 4 26756 -> 56:2 = 28:2 = 14 -> it is divisible by 4 245678 -> 78:2 ⇑ -> is not divisible by 4
• Rules of Divisability: A number is divisible by 5, if if the integer ends in 0 or 5
• Rules of Divisability: A number is divisible by 6, if if the integer is BOTH DIVISIBLE BY 2 AND 3 48 -> 48:3 => 4+8= 12  // 48:2 = EVEN 
• Rules of Divisability: A number is divisible by 8, if If the integer is divisible by 2 THREE TIMES 32: 2= EVEN / 16:2= EVEN // 8:2 = EVEN or if the LAST THREE digits are divisible by 8 2355883556 -> 556 is not divisible by 8
• Rules of Divisability: A number is divisible by 9, if the sum of integers digits is divisible by 9 4185 -> 18 : 3 = 6 -> Divisible
• Rules of Divisability: A number is divisible by 9, if the sum of integers digits is divisible by 9 4185 -> 18 : 3 = 6 -> Divisible
• Rules of Divisability: A number is divisible by 10, if it ends with 0
• Ways to say "3 is a factor of 12" 12 is divisible by 2 12 is a multiple of 3 3 divides 12 3 is a divisor of 12 3 is a factor of 12 12/3 is an integer 12/3 yields a remainder of 0 3 goes into 12 evenly 12 items can be shared among 3 people so that each person has the same number of items 12=3n, where n is an integer
• Divisibility and Addition/Substraction If you add two multiples of 7, you get another multiple of 7 -> (5*7)+(3*7) = (8*7) If you substract two multiples of 7, you get another multiples of 7 -> (5*7)-(3*7) = (2*7) If you add of substract multiples of N, the result is a multiple of N. 
• Primes A prime number is any positive integer larger than 1 with only 2 factors: 1 and itself 1 is not consideres prime, it has only itself MEMORIZE!! : 2,3,5,7,11,13,17,19,23,29
• When to use prime factorization? If a problem states of assumes that a number is an integer, you may need to use prime factorization to solve the problem. Determining whether one number is divisible by another number Determining the greatest common factor of 2 numbers Reducing factors Finding the least common multiple of two or more numbers Simplifying square roots Determining the exponent on one side of an equation with integer constraints.
• Factor foundation rule If a is a factor of b and b is a factor of c, then a is a factor of c example: if 72 is divisible by 12, then 72 is also divisible by all factors of 12 (1,2,3,4,6,12) 72 6     12 3  2  2  2  2  3 
• The prime box Is 27 a factor of 72? -> build a prime box for 72 12       6 2 3 2 2 3 -> 3*2 and 2*3 27 = 3 * 3 * 3  -> 3*3  Therefore, 27 is not a factor of 72
• Given that the integer n is divisible by 3,7 and 11, what other numbers must be divisors of n? 3,7,11 are prime factors of n -> prime box hence, n must also be divisible by all the possible produts of the primes in the box: -> 21, 33, 77 and 231 are other numbers that a divisors of n! prime box for variable: partial prime box prime box for a number: compelte prime box
• Finding Greatest Common Factor and Least Common Multiple Greatest Common Factor: GCF -> the largest divisor of two or more integers -> SMALLER than or equal than Least Common Multiple: LCM -> the smallest multiple of two or more integers -> GREATER than or equal than VENN DIAGRAMS: overlapping or non-overlapping elements Factor the numbers into primes 30= 2*3*5, 24=2*2*2*3 Create a Venn-diagram Place shared numbers into the middle The GCF is the product of primes in the overlapping region: 2*3=6 The LCM is the product of all primes in the diagram: 5*2*3*2*2 = 120
• 3 Ways to express Remainders Dividend = Quotient * Divisor + Remainder Dividend = Multiple of Divisor + Remainder  1. Dividend/Divisor = Quotient + Remainder/Divisor
• When positive integer A is divided by positive integer B, the result is 4.35. Which of the following cold be the remainder when A is divided by B? a/b=4.35 -> this means that 4 is the quotient and 0.35 is the remainder (expressed as a decimal.  0.35 = Remainder / Divisor = R/B  This relationship my not appear 
• Three Ways to Express Remainders Dividend : 17 Divisor : 5 Quotient : 3 Remainder : 2 You can also express this relationship as a general formula: Dividend/Divisor = Quotient + Remainder / Divisor 
• When positive integer A is divided by positive integer B, the result is 4.35. Which of the following could be the remainder when A is divided by B? Formula: Dividend/Divisor = Quotient + Remainder/Divisor  You know that Dividend/Divisor = 4.35   4 is the Quotient 0.35 is the remainder (decimal) 0.35 = Remainder/Divisor = R/B If you let R equal the remainder, set up the following relationship R and B must both be integers 0.35 = R/B 35/100 = R/B 7/20 = R/B   cross-multiply  7B / 20R => divisor must be a multiple of 20, the remainder must be a mulitple of 7. RULE: THE prime factors on the left side of equation must equal prime factors on the right side of the equation
• If s and t are positive integers such that s/t = 64.12, which of the following could be the remainder when s is divided by t? 1. REWRITE: 64.12 t = s 2. SPLIT s = 64 t + 0.12 t 3.) 0.12 t = s-64 t = REMAINDER 4.) Test the answer choices to find a situation in which t will be and integer 0.12 * t = 2 ---> t = 16.67 0.12 * t  = 4 --> t = 33.33 usw. 
• THEORY: How many DIFFERENT and UNIQUE Prime Factors are there? 1. Count the Number of different bases! 
• THEORY: How many TOTAL Prime Factors are there? ( The length of Prime factors) 2. Count the EXPONENTS to get the length of prime factors
• THEORY: How many TOTAL FACTORS are there? (not only primes and 1) n + 1 possibilities 2000 = 2*2*2*2*5*5*5 = 24 * 53 l +1 TOTAL NUMBER OF FACTORS = (4+1) * (3+1) Possibilities Perfect squares have an ODD number of total factors 
• THEORY: PROPERTIES OF PERFECT SQUARES 2. All perfect squares have ONLY EVEN NUMBERS as EXPONENTS 3. All perfect squares have an ODD NUMBER of TOTAL FACTORS  ALL ODD numbers of total factors MUST BE a perfect square 5 factor pairs until it reaches an equal factor pair
• PS STRATEGY: BOXES - If k^3 is divisible by 240, what is the least possible value for k? 240 = 2*2*2*2*3*5 All of k's boxed must be identical  k3 = FILL THE BOXES SO THAT ALL K's ARE IDENTICAL AND ALL SPACES ARE FULL       k      k        k       2      2       2       2      2       2       3      3       5       5      5       5 => DAHER: LEAST POSSIBLE VALUE = 2*2*3*5 = 60
• PS: THEORY: Factorials and Divisibility: N! is a multiple of all integers up to N N! must be divisible by all primes and products of primes in the prime box up to N! 6! = 6*5*4*3*2*1 =  6! is evenly divisible by 5  2,3,4 9! is evenly divisible by 8
• DS: Is p divisible by 168? l) p is divisible by 14 ll) p is divisible by 12 ACTUALY QUESTION: Is P divisible by (168 =) 2*2*2*3*7?  l) P is DIVISIBLE BY 14 => 14= 2*7 NS ll) P is DIVISIBLE BY 12 => 12= 2*2*3 NS TOTAL: l) und ll) 2*2*3*7 NS DO NOT INCLUDE OVERLAPPING NUMBERS !! DIFFERENCE: if it is asked after pq
• DS: Is pq divisible by 168? ACTUALY QUESTION: Is pq divisible by (168 =) 2*2*2*3*7?  l) p is DIVISIBLE BY 14 => 14= 2*7 NS ll) q is DIVISIBLE BY 12 => 12= 2*2*3 NS TOTAL: pq ∈〈2*/*2*3*7SUFFICIENT C Both together are sufficient !! DIFFERENCE: if it is only asked after p
• THEORY: Prime Numbers 2 is the only EVEN prime number 2 is the only number that has no REMAINDER when divided by 2 There is an infinite number of prime numbers Prime numbers do not have a pattern as 2 is the only even prime number PRIME = ALL POSITIVE INTEGERS WITH ONLY 2 FACTORS MUST BE PRIME
• PS: There is no such integer n such that x is divisible by n and 1<n<x !" MEANING:  "there is no such integer n such that x is divisible by.." = X is not divisible by any integer greater 1 and less than x = x can only have 1 and x as factors = x is a prime number 
• THEORY: ADDITION ON MULTIPLES MULTIPLE1 +- MULTIPLE 2 = MULTIPLE 3 MULTIPLE1 +- NON - MULTIPLE 2 = NON-MULTIPLE 3 BUT: MULTIPLE1 +- 2 NON-MULTIPLES 2 = ?  BUT: N=2 => 2 ODDS always sum to an even 
• THEORY: Determining GFC and LCM 1. Determine the factors, e.g. 140=2*5*2*7 2. Create Chart number               2         5       7 100                    22          52       - 140                   22         51       71 250                   21            53       - 3. Look for GFC and LCM  Smallest Count = 21 * 51 * 70 = 10 Highest Cound =  22 * 53 * 73 = 3500
• THEORY: LCM AND FGC relationship M*N = GCF * LCM  Consecutive multiples of n have a GFC of n GFC cannot be larger that (m-n)
• Interpretation: x - y = 4 They are consecutive multiples of 4  Consecutive multiples (x and y) of 4 have a GCF of 4. 
• The dial shown above is divided into 8 equally sized intervals. Ar which of the following letters will the pointer stop of it is rotated clockwise from S through 1174 intervals? There are 8 intervals in each complete revolutions and 1174 intervals. Number of revolutions: 1174 intervals / 8 intervals = 146 R6.  146 complete revolutions 6 intervals The pointer will stop at the 6th interval.
• I fy is an integer, the least possible value of l23-5yl is? 1.) The least possible value = 0 2.) 23 - 5y = 0 when y=4,6 3.) the solution the clostest to 4,6 is 5
• Which of the following expression represents can be written as an integer? (√82 + √82)^2 or (√82 + √82)/82 BOTH: 1. (√82 + √82)2 = ( 2*√82)2 = 4 * 82 is an integer 2.  (√82 * √82)/ 82 = 82/82 = 1
• If K is the sum of the reciprocals of the consecutive integers from 41 to 60 inclusive, which of the following is less than K? The sum 1/41+1/42+1/43....+1/60 has 60-41+1 = 20 fractional terms.  STRATEGY: Look at the maximum and minimum possible values for the sum! MAXIMUM VALUE: largest fraction is 1/41 K is smaller than 20*1/41 => 20/41 = smaller than 1/2 MINIMUM VALUE: smallest fraction is 1/60 K is larger than 20/60 = 1/3 Therefore, 1/2<K<1/3 CHECK FOR ANSWER CHOICES!
• If the range of the 6 numbers is 4,3,14,7,10 and x is 12, what is the difference between the greatest possible value and the least possible value of x? MINIMUM MAXIMUM Analysis Range = Greatest value - Least Value = 12            = 14 - 3 = 11 and not 12 THUS, x must either be 15  or x must either be 2 Difference must be 15-2 = 12                                     
• The sum of all integers k such that -26<k<24 is 1. Eliminate the negatives with positives 1 to 23 -1 to -23               0 2. Add the leftovers : 0 + (-24) + (-25)  = -49
• The closing price of 2420 different stocks on a stock exchange were all different from today's closing price. The number of stocks at a higher closing price was 20% greater than the number that closed at a loser price. How many of the stocks closed at a higher price today? Let n be the number of stocks that closed at a lower price today than yesterday.  H = 1.2 n -> is the number of stocks that closed at a higher price L  =  n 1.2 n + n = 2420 2.2 n = 2420 n= 1320
• Working simultaneously at their respective rates, machine A and B product 800 nails in x hours. Working alone at its constant rate, Machine A produces 800 nails in y hours. In terms of x and y, how many hours does it take Machine B, working alone at its constant rate, to produce 800 nails? RA and RB are the constant rates in nails per hours. Then it follows from the given information that RA+RB = 800/x and RA = 800/y. RB = 800/x + 800/y = 800 * (1/x - 1/y) = 800 (y-x/xy) Therefore xy / y-x hours
• If an integer greater than 6, what must be divisible by 3? QUICKEST:  CHOOSE and integer and eliminate the answer choices 
• A tables gives the age categories of 161 employees at Company X and the number of employees in each category. According to the table, if m is the median age in years of the employees at Company X, then m must satisfy which condition? The median of 161 ages must be the 81st age.  LOOK FOR THAT IN THE TABLE !
• If n is a positive integer and the product of all the integers from 1 to n, inclusive is divisible by 990, what is the least possible value of n? Since N is divisible by 990 every prime factor of 990 must also be a factor of N.  Prima factorization of 990 = 2*3*3*5*11 The least possible value of N with factors of 2,5,3*3 and 11 is 1*2*3*..*11 The least possible value of n is 11. 
• If p is the product of all integers from 1 to 30 (30!), what is the greatest integer k for which 3^k isa factor of p? MAKE A CHART all multiples of 3 between 1 and 30 3,4,8,12,15,18,21,24,27,30  and write down each ones numbers of factors of 3 e.g. 6=1*3 ADD UP THOSE NUMBERS
• Club X has more than 10 but fewer than 40 members. The members sit at tables with 3 members at one and 4 members at each of the other tables. Sometimes the members sit at tables with 3 members at one and 5 members at each of the other tables. If they sat at tables with 6 members at each table except one and fewer than 6 members at that one table, how many members will be at the table that has fewer than 6 members? Dividend/Divisor = Quotient + Remainder Number of Members / Tables = 4 * Tables + 3 Number of Members / Tables = 5 * Tables + 3 Daher:                                      20              + 3 is divisible by 4 and 5 (n-3) = multiple of 4*5 = 20 n-3 = 20 ODER n=23 Of 23 members are divided onto 6 tables, we get 5 leftovers

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