GMAT CLUB

Odd/Even Rules 3 If b is an odd number, which of the following must be even? A. 2𝑏 − 3 B. (𝑏-1)/2 C. (𝑏/2)− 1 D. 𝑏^2 + 2 E. 2𝑏 + 2

Use the "plug numbers" method to check each answer choice. Let's take 1: A. 2*1 - 3 = -1 (Odd) B.(1-1)/2= 0 (Even) - however, just in case, lets try 3. (3-1)/2= 1 (Odd). It usually is a good idea to run through 2 different numbers if you get zero or similar C.(1/2)−1 = −1/2(not an intenger) D. 1 + 2 = 3 (Odd) E. 2 + 2 = 4 (Even)

Factors 2 Do integers usually have an odd or even number of factors?

Vast majority of integers have an even number of factors such as 12 for example: 1, 2, 3, 4, 6, 12. Thus when finding the total number of factors, make sure it is an even number The only integers that have an odd number of factors are perfect squares (that's numbers such as 4, 9, 16, 25, 36, etc). Try 25 for example, it has factors of 1, 5, and 25. Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.

Ratios What is the ratio of a to d if 2a=3b,1/2𝑏=2c, and 3c=d? A)1:2 B)2:1 C)2:3 D)4:3 E)1:5

With questions like these, unless you can spot a shortcut right away, the easiest way to solve is to plug numbers: let's pick a=6, since there is a 2 and 3 involved 2*6 = 3b; b = 4 0.5*4 = 2c; c = 1 3*1 = d; d = 3 Therefore, the ration of a to d is 2:1 The correct answer is B.

Divisibility Rules Is 54780 divisible by 2? Is 1671 divisible by 3? Is 5632 divisible by 4? Is 3830 divisible by 5? Is 2658 divisible by 6? Is 396 divisible by 9?

Yes. Divisibility by 2 - last digit of a number is even Yes. Divisibility by 3 - sum of all digits is a multiple of 3 Yes. Divisibility by 4 - last 2 digits is a multiple of 4 Yes. Divisibility by 5 - the last digit is either 0 or 5 Yes. Divisibility by 6 - the sum of digits is a multiple of 3 and the last digit is even Yes. Divisibility by 9 - the sum of digits is a multiple of 9

Arithmetic to Memorize 7 sqrt 2= sqrt 3= sqrt 625= sqrt 169= List all Primes between 1 and 50 Extra Hard & Extra Credit: 2^6= cbrt 125= 5thrt 243=

sqrt 2= 1.4 sqrt 3= 1.7 sqrt 625= 25 sqrt 169= 13 Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 2^6=64 cbrt 125= 5 5thrt 243= 3

Odd/Even Rules 2 Is 0 odd or even?

0 (zero) is Even It is also not positive or negative- it is neutral

Percent 4 50% of 25 is 25% of which number?

There are 2 solutions to this problem: Long one: 50% of 25 is 12.5 12.5 *100 = x * 25 x = 50 Short one but a harder to come up with: 50% * 25 = x * 25% 50 = x

Number Properties 3 How many distinct integers are there between 1 and 21 inclusive?

Most often GMAT questions will say "inclusive" but sometimes they don't - need to remember to check The formula for calculating the number of integers etween two numbers is: N-M+1 Therefore, the answer is: 21 - 1 + 1 = 21 You can also write out all of the numbers though it s not always possible but just in case: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21

Percent What is the percent formula? How much is 15 percent of 20?

Percent Formula:Part = Percent x Whole For example, 15% of 20 is: 15/100∗20 = 300/100=3

Percent 3 If the production of hybrid cars tripled last year, by how many percent did it increase? a.100% b.200% c.250% d.300% e.333%

The correct answer is B - increased by 200% For example, if production was 10 cars, and it tripled to 30 cars, the increase was 20 cars, which is 200% of 10

Multiples 2 Is 0 (zero) a multiple of 100?

Yes, 0 (zero) is a multiple of everything However, it is unlikely that GMAT will test this property but it helps to remember

Absolute Value 3 What is the value of x If |x - 5| = 10?

|x - 5| = 10 We must evaluate both the positive and negative outcome since the absolute value "removes" the negative sign Thus, we have 2 equations: X - 5 = 10 X - 5 = - 10 (in case it was a negative expression) Thus x = 15 and -5. (Plug in both to check) Absolute value inequalities and equations will almost always have 2 answers/solutions! always have 2 answers/solutions!

Translate The product of three and four is reduced by five and then increased by the difference between the original product and eight = ?

3 x 4 = 12 12 - 5 = 7 12 - 8 = 4 7 + 4 = 11 Answer: 11

Odd/Even Rules Check your knowledge of Odd/Even rules Odd + Odd = ? Odd - Even = ? Even + Odd = ? Odd x Odd = ? Odd ÷ Even = ? Odd x Even = ? Hint: try picking numbers

Odd + Odd = 1+ 1 = 2 (Even) Odd - Even = 3 - 2 = 1 (Odd) Even + Odd = 2 + 1 = 3 (Odd) Odd x Odd = 3 x 3 = 9 (Odd) Odd ÷ Even = 3 ÷ 2 = 1.5 (Not an integer!) Odd x Even = 3 x 2 = 6 (Even)

Powers 3 (2^4x2^8)/(2^6x2)=?

(2^4x2^8)/(2^6x2)= (2^12)/(2^7)= 2^5 Division of powers with the same base (2) is handled similarly to multiplication except the power values are deducted. The base stays unchanged

Powers 4 (3^3)^2=?

(3^3)^2= (3^6) When a number taken to a power is taken to another power, you need to multiply the exponents (2 * 3)

Multiples 4 What is the Least Common Multiple of 18 and 24?

Option 1 Find the factors of 18 and 24: 18 = 3 x 3 x 2 24 = 3 x 2 x 2 x 2 Multiply the unique factors: 3 x 3 x 2 x 2 x 2 = 72 Option 2 Multiply the 2 numbers: 18 x 24 = 432 Shared factors: 2 and 3 Divide 432 by 2 and 3 = 72

Multiples 3 How to find a Least Common Multiple (LCM)?

Option 1: Step 1: Find the prime factors of each of the numbers Step 2: Multiply the unique factors (exclude duplicates) Option 2: Step 1: Multiply the two numbers Step 2: Find any factors the two numbers share Step 3: Divide the product in Step 1 by the factors that the two numbers have in common

Percent 2 What is the formula for percent change? If Jack got a raise from $15 per hour to $18 per hour, what was the percent increase?

Percent Change Formula: 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑐h𝑎𝑛𝑔𝑒/𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡× 100% Jack's Raise: (18 − 15)/15 × 100% = 3/15 × 100% = 100/5 = 20% ! Make sure you use the original amount (15), not the new one (18)

Reciprocal What is a reciprocal?

Reciprocal for a number 𝑥, denoted by 1/x or x^-1, is a number which when multiplied by 𝑥 yields 1. The reciprocal of a fraction a/b is b/a For example reciprocal of 3 is 1/3 Reciprocal of 5/6 is 6/5

Absolute Value 4 Is M<0? −𝑀 = |𝑀| 𝑀2 = 9

Statement (1) by itself is not sufficient. From statement (1), M is either a negative number or zero. If M=-3, then -(-3)=|-3| or 3=3, which is not sufficient. Statement (2) by itself is not sufficient. From statement (2), M can be either 3 or -3, which is not sufficient. Statements (1) and (2) combined are sufficient. If we combine both statements, then M=-3. The correct answer is C.

Odd/Even Rules 4 Is A+B+C even or odd? A - C - B is even (𝐴-𝐶)/𝐵 is odd

Statements (1) and (2) combined are insufficient. Consider A=6, B=4, and C=2 (the answer is "yes") and A=0.5, B=0.3, and C=0.2 (the answer is "no"). Do not assume that the numbers are integers if the question does not mention it. The correct answer is E.

Absolute Value 2 If |x-1| = 4, what is the value of x?

Step 1: Find positive/negative roots: 𝑥 − 1 ≥ 0 in our case, 𝑥 − 1 = 4 𝑥 − 1 < 0 in our case, − 𝑥 − 1 = 4 Step 2: Solve the equations x = 5 x = -3 Step 3: Check conditions (𝑥 − 1 ≥ 0 & 𝑥 − 1 < 0) 5 − 1 = 4 ≥ 0 -3 - 1 = -4 < 0

Absolute Value What is the 3- step approach to solving equations and inequalities with absolute value?

Step1: Open modulus and set conditions. To solve/open a modulus, you need to consider 2 situations to find all roots: Positive (or rather non-negative) Negative Step 2: Solve new equations from Step 1 Step 3: Check conditions for each solution from Step 2

Number Properties 2 If 𝑥^2 = 𝑦 + 5, 𝑦 = 𝑧 − 2, 𝑎𝑛𝑑 𝑧 = 2𝑥, 𝑖𝑠 𝑥^3 + 𝑦^2 + 𝑧 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 7? x > 0 y = 4

Substitute x, y, and z in the original equation to get the following: 𝑥^2 − 2𝑥 − 3 = 0; The solutions to this equation are x=-1 and x=3. Statement 1 is sufficient. If x>0, then the only solution is x=3. The result is 27+16+6 =49, which is divisible by 7 Statement 2 is sufficient. If y=4, plugging in this value into 𝑥^2 = 𝑦 + 5, gives us that x=3 or x=-3. However, based on the above, x=3 or x=-1. Therefore x=3. Sufficient. The correct answer is D.

Percentiles Lena's first test score was at the 80th percentile in a class of 120 students. On another test, 24 out of 200 students scored better than Lena. If nobody had Lena's score, what is Lena's percentile after the two tests?

Take each test result separately: Test 1: 120 x 80th = 96 (she scored better than 96 students) Test 2: She scored better than 176 students (200 - 24) Sum up the results: 96 + 176 = 272 120 + 200 = 320 272/320= 85th percentile

Divisibility 2 Is integer 𝑥2𝑦4 divisible by 9? x is an integer divisible by 3 xy is an integer divisible by 9

The best way to approach this question is to plug in several sets of numbers Many are tempted to plug 3 for x and then for S2, the only value y can have is 3; in that case, the answer is Yes. But if we try x=81 and y=(1/9), then x is an integer divisible by 3, xy is an integer divisible by 9, but 𝑥^2𝑦^4 = 1 and is not divisible by 9. The answer is E

Multiples What is a multiple? Is 5 a multiple of 55 or is 55 a multiple of 5?

The multiple of a number is the product of the number and any other whole number. (For example, 2,4,6,8 are multiples of 2) Therefore 55 is a multiple of 5 For example, 6 has factors 1, 2, 3, 6; and multiples that are 6, 12, 24, 36...

Powers 2^2+2^3+2^4+2= ? A) 2^9+ 2 B) 2^10 C) 2^5+2 D) 2^5 E) 30

There are no rules for adding or subtracting powers. This question is solved by brute force: 4 + 8 + 16 + 2 = 30 The correct answer is (E)

Number Properties Which of the following integers represents a sum of 3 consecutive even integers? 200 303 400 554 570

To answer the question, check which integer is both divisible by 3 (since there are 3 integers) and is even. The only number that falls into both of those 2 categories is 570. Correct answer is E

Arithmetic to Memorize 4 3^2= 3^3= 3^4= 3^5= 4^2= 4^3= 4^4=

3^2=9 3^3=27 3^4=81 3^5=243 4^2=16 4^3=64 4^4=256

Powers 5 3^3^3=?

3^3^3= 3^27 Start operations from outside and work your way in: 3 taken to the power of 3 is 27. Thus the base of 3 is taken to the power of 27.

Powers 8 4^(1/2)= ?

4^(1/2)= sqrt 4= 2 Fraction powers are interpreted as follows: the denominator is the root and numerator is the power. For example: 3^(2/3)= cbrt 3^2 or 2^(1/2)= sqrt 2

Powers 7 5^0= ?

5^0= 1 Any number (positive or negative) taken to the power of 0 (zero) equal to 1. Not tested on the GMAT, zero to the power of zero is also 1.

Arithmetic to Memorize 5 5^2= 5^3= 11^2= 12^2= 13^2= 14^2= 15^2= 16^2=

5^2=25 5^3=125 11^2=121 12^2=144 13^2=169 14^2=196 15^2=225 16^2=256

Powers 2 2^2x2^3x2^4x2^5x2=? A) 2^120 B)32^15 C) 2^15 D) 2^14 E)62

2^2x2^3x2^4x2^5x2= 2^15 When powers with the same base (2 in this case) are multiplied, the powers are summed and the base is held constant. Thus, it is 2 to the power of 2+3+4+5+1=15. The correct answer is (C)

Arithmetic to Memorize 2 1/12= 7/8= 75%= 20%= 16 2/3%= 83 1/3%=

1/12= 8.33% 7/8= 87.5% 75%= 3/4 20%= 1/5 16 2/3%= 1/6 83 1/3%= 5/6

Arithmetic to Memorize 1/2= 1/4= 2/5= 1/20= 1/8= 1/6=

1/2=50% 1/4=25% 2/5=40% 1/20=5% 1/8=12.5% 1/6= 16.67%

Powers 6 2^-2 x 2^-3 x 2^6 x 2^-4= ?

2^-2 x 2^-3 x 2^6 x 2^-4= 2^-3= 1/(2^3)= 1/8 Negative powers turn a number to a reciprocal number taken to that power. For example 2^-2= 1/(2^2) = 1/4

Arithmetic to Memorize 3 2^2= 2^3= 2^4= 2^5= 2^6= 2^7=

2^2=4 2^3=8 2^4=16 2^5=32 2^6=64 2^7=128

Factors How many total factors does 462 have?

Advanced method for finding the number of all possible factors: Write out each prime factor and their power, for example, 12 can be written as follows: 2^2 3^1 5^0 Add 1 to all powers and multiply them: (in this case 6) and that's the number of factors (1, 2, 3, 4, 6, 12) Prime Factors of 462 = 2^1, 3^1, 7^1, 11^1 Total number of factors: 2x2x2x2 =2^4= 16

Arithmetic to Memorize 6 Complete the following: 8, 16, 24, 32, 40, ...... 160 12, 24, 36.... 120 15, 30... 150

Complete the following (Answers): 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120 12, 24, 36, 48, 60, 72, 84, 96, 108, 120 15, 30, 45, 60, 75, 90, 105, 120