GMAT #10 Integer Properties
Divisibility Rules of 9
9 - If the sum of the digits is divisible by 9, so is the number
Evenly spaced numbers
Consecutive integers- always one more than previous Consecutive multiples- separated by same type of multiplier
Only 3 ways to get odd
E + O = O E - O = O E x O = O
Remainders with decimals
For example if a quotient = x/y = 5.12 it can be written as 5 + 12/100, so when trying to find a variable plug values into R/D = N/100
Imaginary Numbers
Imaginary numbers are not logical. They state that any negative number is imaginary because a negative cannot exist under a square root. Perfect square roots must mirror each other. As a result, two positive numbers create a positive result and two negative numbers create a positive result. Thus, a negative number cannot exist under the square root is called an imaginary number
Mean and Median Properties of Evenly Spaced Numbers
(1) The mean always equals the median. (2) The average of the first and last terms equals the mean and the median of the entire set. (3) The mean of a number set with an odd number of terms will ALWAYS be an integer and the mean of a number set with an even number of terms will NEVER be an integer.
Determine total number of factors
(1) determine prime factorization of integer (2) add 1 to each exponent of prime factors (3) multiply the resulting sums Example: 'How many different positive integers are factors of 550?' Answer: 12. The"PF"of550=2^1 × 5^2 × 11^1,since550=10× 55=(2× 5)× (5× 11). Asthe"PF"2^1 × 5^2 × 11^1 contains the exponents 1, 2, and 1, and adding 1 to each of these exponents yields the sums 2, 3, and 2, 550 has 12 total factors,since 2× 3× 2=12
Counting consecutive multiples
(Last - First) / Increment + 1 (I.e wants to know how many multiples of 4 between 20 to 80 inclusive, so subtract and find how many multiples of 4 and add 1) If last number isn't consecutive multiples, round to nearest multiple within in the set
Combining Info to see divisibility
- if seeing SAME variable is divisible by the same prime factors then delete the same number because REDUNDANT - if seeing DIFFERENT variables divisible, then DON'T eliminate the redundancy as it is may NOT be redundant for different variables
Inclusive and Exclusive
- inclusive: "from X to Y", subtract and add one - exclusive: "between X and Y", subtract last and first and subtract one
Evenly spaced set-mean and median
- mean always equals median - average of first and last terms equals the mean and median of entire set - mean of an odd number set will be an integer, mean of an even number set will not be an integer
Length of integer
- number of prime factors - To maximize the length of any integer, always identify the smallest prime factor possible. In most cases, this factor will be 2, since 2 is the smallest prime factor. Then determine the largest power of that factor that is less than or equal to the integer in question to maximize find the smallest prime factor, keep raising power until find the integer closest to the integer asked about (e.g. 256, 2^8 = 256 for number less than 300) - if value of integer is odd, use 3 as smallest prime factorization instead of 2 since 3 is the smallest odd prime factor
Statements that indicate prime numbers
- there is no integer p such that q is divisible by p and 1 < p < q - Integer q has q distinct factors, and q > 1. (q = 2) - The difference of any two distinct positive factors of n is odd. (n = 2) so it's 2 because needs to be even to be divisible by 2 - Integer p cannot be expressed as the product of two integers, each of which is greater than 1. (p is prime)
"Zero's" Properties
0 is an integer, even, n +-0= n, n x 0 = 0, and division by 0 is impossible = undefined 0 is a multiple of every integer
Prime Numbers
1 is not a prime number since 1 only has 1 prime factor. Prime numbers only have 1 and the number itself 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37
Multiplication chains
1. Multiplication chains are divisible by each of the integers that they contain (7 × 11 × 13)/ 13= 7 × 11 = 77 2. Multiplication chains are divisible by the factors of the integers that they contain. (11×12×13)/6=11× 2×13=286 3. Multiplication chains are divisible by any integer that is added or subtracted from them, as long as that number or its factors are contained within the chain! For example, the sum of 5! + 4 is divisible by 2 and 4, since 5! and 4 are both divisible by 2 and 4. Likewise, the difference of 8! - 14 is divisible by 2, 7, and 14, since 8! and 14 are both divisible by 2, 7, and 14. Note, conversely, that the sum of 8! + 13 may be prime, as 8! and 13 share no common factors. 4. The smallest prime factor of any multiplication chain to which 1 has been added must be larger than the largest prime number in the chain!
Divisibility Rules of 10
10 - If the number ends in 0, it is divisible by 10.
Divisibility Rules of 2
2 - If the last digit is even, the number is divisible by 2.
Divisibility Rules of 3
3 - If the sum of the digits is divisible by 3, the number is also.
Divisibility Rules of 4
4 - If the last two digits form a number divisible by 4, the number is also divisible by 4 If the last two digits of a number can be cut in half twice then divisible by 4
Divisibility Rules of 5
5 - If the integer ends in 0 or 5
Divisibility Rules of 6
6 - If the number is divisible by both 2 and 3, it is also divisible by 6.
Divisibility Rules of 8
8 - If the last three digits of a number are divisible by 8, then so is the whole number (if the last three digits can be cut in half thrice)
Perfect Cubes and Divisibility
A PERFECT CUBE is the cube of an integer and contains prime factors whose EXPONENTS are ONLY 3 A NON-PERFECT CUBE contains one more prime factors with EXPONENT that is NOT 3 To get perfect cube increase exponent until all MULTIPLES OF 3
Perfect Squares and Divisibility
A PERFECT SQUARE is the square of an integer and contains prime factors whose exponents are ONLY EVEN A NON-PREFECT SQUARE contains one more prime factors with exponent that is ODD. To get perfect square increase exponent until all even.
Identifying Factors
Approximate the square root of that integer and then apply the divisibility rules (e.g. for 80, less than 9^2 so test numbers 1 - 8.
Consecutive multiples
Consecutive multiples are integers that are divisible by the gaps between them. For example, the number set {4, 8, 12, 16, 20} could also be referred to as the set of consecutive multiples of 4 from 4 to 20, since each term in the set is divisible by the gap of 4 that separates each term.
Divisibility and Prime Factorization
Divisibility is tied to multiples and factors For example, 6 is divisible by 2, since the division of 6 and 2 yields an integer. Note, however, that 2 is also a factor of 6, and that 6 is a multiple of 2. If all factors in denominator are in the numerator, it is divisible 42 = (2 x 3 x 7) / (2x3) = 7
Finding greatest common factor of large numbers
Do chart for each number and find prime factorization. For the prime factors in common, find the products of the prime factors in COMMON The largest factor shared by two or more integers is traditionally referred to as the Greatest Common Factor.
Addition - odd/ even rules
E + E = E O + O = E E + O = O
Subtraction - odd/even rules
E - E = E O - O = E E - O = O
Multiplication - odd/even rules
E x E = E O x O = O E x O = E
Division. - odd and even rules
E/E = e, o, fraction E/O = e, fraction O/O = o, fraction O/E = fraction
Odd/Even
Even divisible by 2, odd not divisible by 2 (2n + 1) 0 is considered even decimals and fractions aren't considered even or odd
Multiples - "M and M rule"
Every integer has 'More Multiples than factors'
Factors
Factors of an integer are integers that evenly divide INTO that integer Only positive numbers can be factors If a and b are integers (with a not zero), we say a divides b if there is an integer c such that b = ac.
Remainder 'Proper Form'
I = D x Q + R (dividend = divisor x quotient + remainder) While it may be unexpected, it is worth noting that any instance of division that has a quotient of zero still has a remainder! For example, 2 ÷ 7 has a remainder of 2, even though it has no quotient, since 7 goes into 0 zero times but leaves 2 "left over" when it does.
Sum and Difference Rule - Multiples
If I = an integer, the sum and the difference of any two multiples of I are ALWAYS a multiple of I as well For example, the sum and difference of 49 and 14 are both multiples of 7, since 49 and 14 are each multiples of 7. Thus, 49 + 14 =63 (7 × 9) and 49 - 14 = 35 (7 × 5).
Divisibility - Addition Rule (Multiples)
If have ODD number of CONSECUTIVE integers the sum will always be a multiple of the number of integers If have EVEN number of CONSECUTIVE integers the sum will NOT be a multiple of the number of integers
Divisibility when adding integers to "x"
Manhattan Prep DS Question Is x a multiple of 4? if x + 2 is divisible by 2, then x itself must be divisible by 2, but not necessarily 4. (RULE: for x + y to be divisible by y, x itself must be divisible by y)
Divisibility Concepts
More advanced decimal questions often involve divisibility concepts. To handle such questions, let R/D = N/100 and cross-multiply. Each variable will be a multiple of the integer on the other side of the equation!
Adding evenly spaced integers
Number of terms X average = sum
Range of Remainders
Remainders must always be bigger than the divisor and can't be negative
Rewriting decimal problems
Rewriting decimals in the form Q + N/100 is extremely useful for remainder problems, as it allows us to directly equate N/100 to R/D as N/100 represents the remainder portion of a numeral in decimal notation and R/D represents the remainder portion of a numeral in fractional notation.
dividend vs divisor
Technically, the integer that gets divided is known as the dividend and the integer that does the dividing is known as the divisor.
Prime Factors
The factors of a number that are prime numbers. Every integer consists of 1 or more prime factors. Prime numbers only have one prime factor since 1 is NOT prime
Finding least common multiple of large numbers
The least common multiple of two or more numbers is the smallest integer divisible by those integers. Do chart of prime factorization then multiply the greatest prime factors of each integer together - doesn't have to be in COMMON
Counting odds and evens in Consecutive Multiples
To count evens and odds treat them like multiples of 2. If inclusive then need to add 1 if start and end on even or odd add one more to count
Integers
Whole numbers include 0 and neg. numbers Every integer is a factor and multiple of itself
Remainders
a part, number, or quantity that is left over including if 2 is divided by 7, since 7 goes into 2 0 times, it has a remainder of 2
Finding integers with remainders
a quick way to generate ONE example of an integer that leaves a remainder R when divided by D is to add R and D! For example, an example of an integer that leaves a remainder of 5 when divided by 6 is 11, as 5 + 6 = 11.
Identifying Prime Numbers
approximate its square root and test the relevant divisibility rules.
Divisibility rule of 11
complex way to test divisibility by 11 is to assign opposite signs to adjacent digits and then to add them to see if they add up to 0. For example, we know that 121 is divisible by 11 because -1 +2 -1 equals zero. In our case, the two 9s, when assigned opposite signs, will add up to zero, and so will the digits of 440, since +4 -4 +0 equals zero
Sum and Difference Rule - Not Multiples
if I = an integer, the sum and difference of a multiple of I and a non-multiple of I are NEVER multiples of I. For example, the sum and difference of 49 and 20 are NOT multiples of 7, since 49 is a multiple of 7 and 20 is not a multiple of 7. Thus, 49 + 20 = 69 and 49 - 20 = 19.
Remainders with variables
put in proper form to solve and find value of variables I = D x Q + R (dividend = divisor x quotient + remainder)
Multiples
the multiples of a number are divisible by that number without a remainder (e.g. multiples of 5 are 5, 10, 15, 20...) multiples can be negative
Divisibility - Multiplication Rule
the product of N evenly spaced terms is always a multiple of N! to determine the factors of N evenly spaced terms, substitute the smallest possible values for those N terms and multiply to see if it can be a multiple (e.g. if abc consecutive multiples of 2, is abc a mutliple of 24? 2x3x4 = 12, so yes.