GRE Quantitative (Arithmetic) exam Cheat sheet

Integers

• Integers : ..-3,-2,-1,0,1,2,...
  
• A whole number either positive, negative or zero. Example: (-1.-2.-3 .0,1,2,3,4…….)
• If an integer is divisible by 2, it is called an even integer; otherwise it is an odd integer. Note that when a positive odd integer is divided by 2, the remainder is always 1. The set of even integers is ..., 6, 4, 2,0,2,4,6, ... and the set of odd, integers is ..., -5, -3, -1,1,3,5, ...
• The sum of two odd integers is an even integer.
• The sum of an even integer and an odd integer is an odd integer.
• The product of two odd integers is an odd integer.
• The product of an even integer and an odd integer is an even integer
• The product of two even integers is an even integer.
• 0 is neither neither positive or negative. 0 is even number.
• real numbers are integers and have decimal values between them..
• a⁰=1, 1/aⁿ=a^-n
• The absolute value of a number N, denoted by N , is defined to be N if N is positive or zero and -N if N is negative. Example |-5| =5
• Integer divisible by 2 is called even integer (including 0 and 2). Examples are -8,-6,-4,-2,0,2,4,6,8

Factors

• Factors of a number are it's divisors. For instance, the factors of 21 are 3 and 7, because 3*7 = 21.
• The greatest common divisor (or greatest common factor) of two nonzero integers x and y is the greatest positive integer that is a divisor of both x and y. For example GCF of 30 and 45 is 15, GCF of 60 and 90 is 30.
• Least Common Multiple: (LCM) of two non â€"zero integers x and y is the smallest +ve integer that is a divisor of both x and y, for example LCM of 50 and 60 is 300, LCM of 30 and 40 is 120.
• A prime number is an integer greater than 1 that has only two positive divisors: 1 and itself, examples are 2,3,5,7,11,13,17 (remember 2 is a prime number). 1 is not a prime number, 2 is the smallest prime and the only even prime. Primes numbers below 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 ,61,67,71,73,79,83,97

Divisibility

• 3: sum of digits divisible by 3
• 4: the last two digits of number are divisible by 4
• 5: the last digit is either a 5 or zero
• 6: even number and sum of digits is divisible by 3
• 8: if the last three digits are divisible by 8
• 9: sum of digits is divisible by 9, 11: A Number is divisible by 11 when diff of [sum of all alternate number starting from unit’s place sum] AND [sum of all remaining number is divisible] by 11. e.g.,. 46816 (6+8+4) -(1+6) = 11 hence divisible by 11

RATIONAL NUMBERS

All integers belong to the rational numbers. A rational number is a number which can be written as a/b, b≠0, where both a and b are integers, for example: number 7 is an integer as well as a rational number, it is an integer as it can be written without a decimal component and a rational number as it can be written as 7/1, whereas 1/4 =0.25 is a rational number but not an integer, or repeating as in 1/6 = 0.8333. All rational numbers belong to the real numbers

IRRATIONAL NUMBERS

Every terminating or repeating decimal represents a rational number, not all decimals are terminating or repeating; for instance, the decimal that is equivalent to √3 is 1.7320508075-----and it can be shown that this decimal does not terminate or repeat. Since this decimal does not terminate or repeat, they are not rational numbers. Such numbers are called irrational numbers

ABSOLUTE VALUE

A number's absolute value is the distance between a number x and 0. It is represented as |X|. The absolute value of a number X, denoted by |X|, is defined to be X, if X is positive or zero and -X if X is negative. Example |-5| = 5 For any number a, and positive number b, (a) |a|=b gives a=b or a=-b (b) |a| < b gives -b < a < b (c) |a| >b gives a < -b or a > b

• Reciprocal of a number between 0 and 1 (e.g. 1/3, 2/5) is greater than original number. if 0x, Also 1/x >1 for ex. 1/0.2 =5 >1>0.2
• Multiplying a positive number by a fraction between 0 and 1 gives a smaller number than original If 0Square of a number between 0 and 1 is smaller than the original number. If 0 < x < 1, and a and b are integers with a>b>1, then xa< xb for ex. (1/2)⁵< (1/2)² < (1/2)
• Square root of a number between 0 and 1 is greater than the original number. If 0 < x < 1, then √x >x, √ (3/4)>3/4,

RATIO

A ratio can be written in 3 different ways and can be read as the ratio of x to y or x: y or x/y If a set of objects is divided into 2 groups in the ratio of x: y, then the first group is a/(a+b) and the second group contains b/(a+b)

PROPORTION

Proportion is an equation where two ratios are equivalent for eg: if one pack of flour mix gives us 20 cakes, then 2 pack will give us 40 cakes, 40 /2 = 20/ 1, A proportion can be read as if a is to b as c is to d, and can be written as a/b =c/d where b, d≠0. If one number in proportion is unknown, we can find that number by solving the proportion.

PERCENT

Percent = Part / Whole. The term percent means per hundred; Percent’s are ratios that are represented as parts of a whole. The easiest way to convert a fraction to a percentage is to divide the numerator by denominator then multiple the result by 100, and then add % sign. And to convert a percentage to a fraction, we must first convert it to a decimal (divide by 100) and then use the steps of converting decimal to fractions for example :(a)15 is 75% of x, find number x? Convert the math into English, per above problem 75/100 of a number is 15, (75/100) x=15, gives x =

Some real problems...

15 is 77% of some number x, find number x?
First thing is to form a equation for above problem, according to above problem 60/100 of number is 15, which means
(75/100)x=15, which means x = 20.