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FINDING THE NUMBER OF FACTORS OF AN INTEGER

First make prime factorization of an integer n=a^p*b^q*c^r, where a, b, and c are prime factors of n and p, q, and r are their powers. The number of factors of n will be expressed by the formula (p+1)(q+1)(r+1). NOTE: this will include 1 and n itself. Example: Finding the number of all factors of 450: 450=2^1*3^2*5^2 Total number of factors of 450 including 1 and 450 itself is (1+1)*(2+1)*(2+1)=2*3*3=18 factors.

Triangle Property

For triangles with same area, the perimeter is smallest for an equilateral triangle. For triangles with same perimeter, the area is maximum for an equilateral triangle. (If you think about it, this property goes hand in hand with the one we used in St. 2).

Co-prime

Two consecutive integers are co-prime, which means that they do not share any common factor other than 1.

The only positive integer which has only 1 even factor is 2

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Zero

1. 0 is an integer. 2. 0 is an even integer. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder and as zero is evenly divisible by 2 then it must be even. 3. 0 is neither positive nor negative integer (the only one of this kind). 4. 0 is divisible by EVERY integer except 0 itself, (or, which is the same, zero is a multiple of every integer except zero itself).

Remainders

1. 0\leq{r}<x means that remainder is a non-negative integer and always less than divisor. 2. When y is divided by x the remainder is 0 if y is a multiple of x. For example, 12 divided by 3 yields the remainder of 0 since 12 is a multiple of 3 and 12=3*4+0. 3. When a smaller integer is divided by a larger integer, the quotient is 0 and the remainder is the smaller integer. For example, 7 divided by 11 has the quotient 0 and the remainder 7 since 7=11*0+7 4. The possible remainders when positive integer y is divided by positive integer x can range from 0 to x-1. For example, possible remainders when positive integer y is divided by 5 can range from 0 (when y is a multiple of 5) to 4 (when y is one less than a multiple of 5).. 5. If a number is divided by 10, its remainder is the last digit of that number. If it is divided by 100 then the remainder is the last two digits and so on. For example, 123 divided by 10 has the remainder 3 and 123 divided by 100 has the remainder of 23.

Prime Numbers

1. 1 is not a prime, since it only has one divisor, namely 1. 2. Only positive numbers can be primes. 3. There are infinitely many prime numbers. 4. the only even prime number is 2. Also 2 is the smallest prime. 5. All prime numbers except 2 and 5 end in 1, 3, 7 or 9.

Absolute values

1. |x|≥0 2. √x²=|x| 3. |0|=0 4. |-x|=|x| 5. |x-y|=|y-x|. |x - y| represents the distance between x and y, so naturally it equals to |y - x|, which is the distance between y and x. 6. |x|+|y|≥|x+y|. Note that "=" sign holds for xy≥{0} (or simply when x and y have the same sign). So, the strict inequality (>) holds when xy<0; 7. |x|-|y|≤ |x-y|. Note that "=" sign holds for xy>{0} (so when x and y have the same sign) and |x|≥|y| (simultaneously).

Divisibilty Rules

1. 2 - If the last digit is even, the number is divisible by 2. 2. 3 - If the sum of the digits is divisible by 3, the number is also. 3. 4 - If the last two digits form a number divisible by 4, the number is also. 4. 5 - If the last digit is a 5 or a 0, the number is divisible by 5. 5. 6 - If the number is divisible by both 3 and 2, it is also divisible by 6. 6. 7 - Take the last digit, double it, and subtract it from the rest of the number, if the answer is divisible by 7 (including 0), then the number is divisible by 7. 7. 8 - If the last three digits of a number are divisible by 8, then so is the whole number. 8. 9 - If the sum of the digits is divisible by 9, so is the number. 9. 10 - If the number ends in 0, it is divisible by 10. 10. 11 - If you sum every second digit and then subtract all other digits and the answer is: 0, or is divisible by 11, then the number is divisible by 11. Example: to see whether 9,488,699 is divisible by 11, sum every second digit: 4+8+9=21, then subtract the sum of other digits: 21-(9+8+6+9)=-11, -11 is divisible by 11, hence 9,488,699 is divisible by 11. 11. 12 - If the number is divisible by both 3 and 4, it is also divisible by 12. 12. 25 - Numbers ending with 00, 25, 50, or 75 represent numbers divisible by 25.

Factors

1. A divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. In general, it is said m is a factor of n, for non-zero integers m and n, if there exists an integer k such that n = km. 2. 1 (and -1) are divisors of every integer. 3. Every integer is a divisor of itself. 4. Every integer is a divisor of 0, except, by convention, 0 itself. 5. Numbers divisible by 2 are called even and numbers not divisible by 2 are called odd.

Even/Odd

1. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder. 2. An odd number is an integer that is not evenly divisible by 2. 3. According to the above both negative and positive integers can be even or odd.

Irrational Numbers

1. An irrational number is any real number that cannot be expressed as a ratio of integers. 2. The square root of any positive integer is either an integer or an irrational number. So, \sqrt{x}=\sqrt{integer} cannot be a fraction, for example it cannot equal to 1/2, 3/7, 19/2, ... It MUST be an integer (0, 1, 2, 3, ...) or irrational number (for example \sqrt{2}, \sqrt{3}, \sqrt{17}, ...).

Divisibility

1. Every GMAT divisibility question will tell you in advance that any unknowns represent positive integers (ALL GMAT divisibility questions are limited to positive integers only). 2. On the GMAT when we are told that a is divisible by b (or which is the same: "a is multiple of b", or "b is a factor of a"), we can say that: (i) a is an integer; (ii) b is an integer; (iii) \frac{a}{b}=integer

WHEN THE SUM OR THE DIFFERENCE OF NUMBERS IS A MULTIPLE OF AN INTEGER

1. If integers a and b are both multiples of some integer k>1 (divisible by k), then their sum and difference will also be a multiple of k (divisible by k): Example: a=6 and b=9, both divisible by 3 ---> a+b=15 and a-b=-3, again both divisible by 3. 2. If out of integers a and b one is a multiple of some integer k>1 and another is not, then their sum and difference will NOT be a multiple of k (divisible by k): Example: a=6, divisible by 3 and b=5, not divisible by 3 ---> a+b=11 and a-b=1, neither is divisible by 3. 3. If integers a and b both are NOT multiples of some integer k>1 (divisible by k), then their sum and difference may or may not be a multiple of k (divisible by k): Example: a=5 and b=4, neither is divisible by 3 ---> a+b=9, is divisible by 3 and a-b=1, is not divisible by 3; OR: a=6 and b=3, neither is divisible by 5 ---> a+b=9 and a-b=3, neither is divisible by 5; OR: a=2 and b=2, neither is divisible by 4 ---> a+b=4 and a-b=0, both are divisible by 4.

HCF and LCM

1. The greatest common divisor (GCD), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder. So GCD can only be positive integer. It should be obvious as greatest factor of two integers can not be negative. For example if -3 is a factor of two integer then 3 is also a factor of these two integers. 2. The lowest common multiple (LCM), of two integers a and b is the smallest positive integer that is a multiple both of a and of b. So LCM can only be positive integer. It's also quite obvious as if we don not limit LCM to positive integer then LCM won't make sense any more. For example what would be the lowest common multiple of 2 and 3 if LCM could be negative? There is no answer to this question. 3. Divisor of a positive integer cannot be more than that integer (for example 4 doesn't have a divisor more than 4, the largest divisor it has is 4 itself). From this it follows that the greatest common divisor of two positive integers x and y can not be more than x or y.

Perfect Squares

1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square; 2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50; 3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors); 4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: 36=2^2*3^2, powers of prime factors 2 and 3 are even.

Inequalities

1. You can only add inequalities when their signs are in the same direction: If a>b and c>d (signs in same direction: > and >) --> a+c>b+d. Example: 3<4 and 2<5 --> 3+2<4+5. 2. You can only apply subtraction when their signs are in the opposite directions: If a>b and c<d (signs in opposite direction: > and <) --> a-c>b-d (take the sign of the inequality you subtract from). Example: 3<4 and 5>1 --> 3-5<4-1. 1. We can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality). For example: 2<4 --> we can square both sides and write: 2^2<4^2; 0<={x}<{y} --> we can square both sides and write: x^2<y^2; But if either of side is negative then raising to even power doesn't always work. For example: 1>-2 if we square we'll get 1>4 which is not right. So if given that x>y then we can not square both sides and write x^2>y^2 if we are not certain that both x and y are non-negative. 2. We can always raise both parts of an inequality to an odd power (the same for taking an odd root of both sides of an inequality). For example: -2<-1 --> we can raise both sides to third power and write: -2^3=-8<-1=-1^3 or -5<1 --> -5^2=-125<1=1^3; x<y --> we can raise both sides to third power and write: x^3<y^3. 1. If both sides of both inequalities are positive and the inequalities have the same sign, you can multiply them. For example, for positive x, y, a, b, if x < a and y < b, then xy < ab. 2. If both sides of both inequalities are positive and the signs of the inequality are opposite, then you can divide them. For example, for positive x, y, a, b, if x < a and y > b, then {x}/{y} < {a}/{b} (The final inequality takes the sign of the numerator). 1. Whenever you multiply or divide an inequality by a positive number, you must keep the inequality sign. 2. Whenever you multiply or divide an inequality by a negative number, you must flip the inequality sign. 3. Never multiply (or reduce) an inequality by a variable (or the expression with a variable) if you don't know the sign of it or are not certain that variable (or the expression with a variable) doesn't equal to zero.

Rule for xⁿ-yⁿ

Always divisible by x-y If n is even , divisible by x+y

Exponents

a^m^n=a^{(m^n)} and not (a^m)^n (if exponentiation is indicated by stacked symbols, the rule is to work from the top down)

Percent Change Formula

if there are two succesive changes to a quantity ( as in this case ) you can solve it by a simple formula net change = a + b + {ab}/{100} e.g. net change of to increase of 50% = 50 + 50 + {2500}/{100} = 125 net change of increase in 8% and decrease of 5 % = 8 - 5 -{8*5}/{100} = 2.6

Roots

nth Root of any positive integer greater than 1 is always greater than 1.