Reciprocal
The reciprocal or the inverse of a fraction is that fraction flipped over; top to bottom and bottom to top.
i.e., the reciprocal of 2/5 is 5/2.
the reciprocal of 13 is 13/1 flipped over, hence 1/13
reciprocal doesn't change signs
i.e., the reciprocal of 2/5 is 5/2.
the reciprocal of 13 is 13/1 flipped over, hence 1/13
reciprocal doesn't change signs
1/2 is the same as 2/4
In both fractions the part represents one-half of the whole. - the fraction was expanded
Expanding fractions
In any fraction you can multiply the numerator and denominator by the same number, without changing the relationship between them.
expanding 3/4 by 5 would yield 15/20
expanding 3/4 by 5 would yield 15/20
reducing fractions
you can divide the numerator and denominator of any fraction by the same number.
reducing 14/42 by 2 would yield 7/21. Keep reducing by 7 to get 1/3.
reducing 14/42 by 2 would yield 7/21. Keep reducing by 7 to get 1/3.
32 x 4 to a-2 =?
32 x 4 to a-2 =?
Start with 4 to a-2 = 4 to a / 4 to 2 = 4 to a / 16.
32 x 4 to a / 16 = 2 x 4 to a or 1 x 2 to a
Start with 4 to a-2 = 4 to a / 4 to 2 = 4 to a / 16.
32 x 4 to a / 16 = 2 x 4 to a or 1 x 2 to a
Adding and Subtracting Fractions
To add/subtract fractions with the same denominators (bottoms), simply add/subtract the numerators (ups).
To quickly add/subtract fractions with different denominators use the bowtie. For example, 1/6 + 1/8 will work out as:
To quickly add/subtract fractions with different denominators use the bowtie. For example, 1/6 + 1/8 will work out as:
Multiplying Fractions
To multiply fractions, multiply straight across, tops with tops, bottoms with bottoms.
If possible reduce before you multiply.
When multiplying a fraction and an integer, first write the integer as a fraction, then multiply.
12/13 x 32/48
reduce 12 & 48 ... 13 & 32
1/1 x 3/4
3/4
3/8 x 24 = 3/8 x 24/1
reduct 8 & 24 = 3/1 x 3/1 = 9/1 = 9
If possible reduce before you multiply.
When multiplying a fraction and an integer, first write the integer as a fraction, then multiply.
12/13 x 32/48
reduce 12 & 48 ... 13 & 32
1/1 x 3/4
3/4
3/8 x 24 = 3/8 x 24/1
reduct 8 & 24 = 3/1 x 3/1 = 9/1 = 9
Dividing Fractions
To divide one fraction by the other, find the reciprocal of the second fraction (the divisor), then multiply.
Remember: Dividing by a fraction is the same as multiplying by its reciprocal.
When dividing a fraction and an integer, first write the integer as a fraction, then divide.
1/4 : 4 = 1/4 x 1/4 = 1/16
Remember: Dividing by a fraction is the same as multiplying by its reciprocal.
When dividing a fraction and an integer, first write the integer as a fraction, then divide.
1/4 : 4 = 1/4 x 1/4 = 1/16
Use Bowtie to compare problems
Is 5/6 greater than 6/7 ?
use the X portion of bow tie. Multiply one denominator by the others multipler to determine which is greater. Don't do the bottom portion of the bowtie (multiplying the denominators.
7x5= 35 6x6=36 thus 5/6 is less than 6/7
use the X portion of bow tie. Multiply one denominator by the others multipler to determine which is greater. Don't do the bottom portion of the bowtie (multiplying the denominators.
7x5= 35 6x6=36 thus 5/6 is less than 6/7
Which is greater 3/15 or 3/16
The denominator of a fraction has a counter-intuitive effect on the fraction's value:
The greater the denominator, the smaller the fraction (under the same numerator).
3/15 is greater than 3/16
The greater the denominator, the smaller the fraction (under the same numerator).
3/15 is greater than 3/16
Which is greater 1/10 or 1/5
The smaller the denominator, the greater the fraction (under the same numerator).
1/10 is less than 1/5
1/10 is less than 1/5
denominator dynamics:
for a fraction to grow greater, increase the numerator and decrease the denominator.
Division Remainder
Remainder is defined as the distance (in units) from the dividend to the nearest multiple of the divisor that is smaller than the dividend.
when 8 is divided by 3, the remainder - the distance between 8 and the nearest multiple of 3 (which is 6) - is 2.
The remainder can never be equal to, or greater than the divisor.
when 8 is divided by 3, the remainder - the distance between 8 and the nearest multiple of 3 (which is 6) - is 2.
The remainder can never be equal to, or greater than the divisor.
When dividing by 5, the highest possible remainder is ?
4
When X is divided by 5, the remainder is 3 what do you plug in?
the remainder 3
when plugging in is difficult to use, use the following formula: for any integer i divided by another integer d
i = quotient·d + remainder
Thus, if dividing i by 5 leaves a remainder of 3, i can be expressed as the equation:
i=5x+3
x being the quotient.
when plugging in is difficult to use, use the following formula: for any integer i divided by another integer d
i = quotient·d + remainder
Thus, if dividing i by 5 leaves a remainder of 3, i can be expressed as the equation:
i=5x+3
x being the quotient.
if x is divisible by 5, is there a remainder?
if x is divisible by 5, this means that x is divisible by 5 with no remainder. Note that this also means that x is a multiple of 5 - since there's no distance between x and the nearest multiple of 5.
What is a factorial?
Factorials are a necessary concept for dealing with Combinations and Permutations problems.
Factorials are marked by a "!" sign.
n! means "n multiplied by all consecutive integers less then n down to 1": n·(n-1)·(n-2)·...·1.
For example: 5! = 5·4·3·2·1
Factorials are marked by a "!" sign.
n! means "n multiplied by all consecutive integers less then n down to 1": n·(n-1)·(n-2)·...·1.
For example: 5! = 5·4·3·2·1
0!
0! = 1! = 1
What is .6 as a fraction?
6/10 or reduce to 3/5
3/5 = .6 duh!
3/5 = .6 duh!
for power numbers Even integer exponent means the result is even or odd?
Even:
2 to 2 = 4 (even)
3 to 2 = 9 (odd)
for powers (with a positive integer exponent), the base rules . If the base is even, the result is even. If the base is odd, the result is odd.
2 to 2 = 4 (even)
3 to 2 = 9 (odd)
for powers (with a positive integer exponent), the base rules . If the base is even, the result is even. If the base is odd, the result is odd.
What are the odd/even rules for division?
when dividing Even and Odd numbers, there are no rules. The result could be even, odd or even a fraction - depending on the numbers.
However, an odd number divided by an even number will result in a fraction.
An odd number is not divisible by 2. An even number is divisible by 2, and will therefore be a multiple of 2. Therefore, an odd integer will never be divisible by an even integer, and the result will be a fraction..
However, an odd number divided by an even number will result in a fraction.
An odd number is not divisible by 2. An even number is divisible by 2, and will therefore be a multiple of 2. Therefore, an odd integer will never be divisible by an even integer, and the result will be a fraction..
Even integer divided by an even integer = even answer?
when dividing Even and Odd numbers, there are no rules. The result could be even, off or a fraction.
Some GMAT integers questions rely on this form of linear thinking to trick careless and hasty test-takers. Remember this rule, and do not be fooled.
Some GMAT integers questions rely on this form of linear thinking to trick careless and hasty test-takers. Remember this rule, and do not be fooled.
Even/odd rules for addition/subtraction
adding / subtracting even and odd numbers:
Even ± Even = Even (e.g. 2+2=4; 4-2=2)
Even ± Odd = Odd (e.g. 2+1=3; 2-1=1)
Odd ± Odd = Even (e.g. 1+1=2; 3-1=2)
Even ± Even = Even (e.g. 2+2=4; 4-2=2)
Even ± Odd = Odd (e.g. 2+1=3; 2-1=1)
Odd ± Odd = Even (e.g. 1+1=2; 3-1=2)
Factor
Is always positive.
A factor is a positive integer that divides evenly into another integer.
A factor of n is a positive integer that n is divisible by with no remainder.
So if 6 is divisible by 1, 2, 3 and 6, these positive integers are all factors of 6.
A factor is a positive integer that divides evenly into another integer.
A factor of n is a positive integer that n is divisible by with no remainder.
So if 6 is divisible by 1, 2, 3 and 6, these positive integers are all factors of 6.
Factor & Divisor
Remember this: a Factor is a positive divisor.
So if 6 is divisible by 1, 2, 3 and 6, these positive integers are all factors of 6.
In the above example, 1, 2, 3, and 6 are all divisors, as well as factors, of 6.
So if 6 is divisible by 1, 2, 3 and 6, these positive integers are all factors of 6.
In the above example, 1, 2, 3, and 6 are all divisors, as well as factors, of 6.
5 is a factor of x. Does x/5 = an integer?
If 5 is a factor of x, then
x is divisible by 5 , and therefore x/5
must give an integer result without remainder.
x is divisible by 5 , and therefore x/5
must give an integer result without remainder.
Even/Odd rules for multiplication/division
Even × Even = Even (e.g. 2×2=4)
Even × Odd = Even (e.g. 2×3=6)
Odd × Odd = Odd (e.g. 1×1=1)
Even times any other integer (odd or even) always comes out even - the result will always be divisible by 2.
The only way for the product of two numbers to be odd is if both of those numbers are odd.
Even × Odd = Even (e.g. 2×3=6)
Odd × Odd = Odd (e.g. 1×1=1)
Even times any other integer (odd or even) always comes out even - the result will always be divisible by 2.
The only way for the product of two numbers to be odd is if both of those numbers are odd.
Factors = multiples?
NO... Factors and multiples are essentially opposite terms:
Factors of a number are positive integers that the number divides into.
Multiples of a number are formed by multiplying that number by any integer.
Factors of a number are positive integers that the number divides into.
Multiples of a number are formed by multiplying that number by any integer.
All of these phrases mean the same thing:
all of these GMAT question phrasings mean the same thing:
1) 5 is a factor of x
2) 5 is a divisor of x
3) x is divisible by 5
4) x/5 is an integer
5) x = 5 times integer
6) x is a multiple of 5
All of the above basically tell you that x=5, 10, 15, 20, 25...
1) 5 is a factor of x
2) 5 is a divisor of x
3) x is divisible by 5
4) x/5 is an integer
5) x = 5 times integer
6) x is a multiple of 5
All of the above basically tell you that x=5, 10, 15, 20, 25...
Is 0 a factor of another integer?
NO.
Zero is never a factor of another integer, as dividing by zero is not defined.
Zero is never a factor of another integer, as dividing by zero is not defined.
Is 0 a multiple of an integer?
YES.
Zero is a multiple of any integer. For example, zero is a multiple of five, as it is basically 5⋅0 - which is still 5 times an integer.
Thus, if x is a multiple of 5, then x could still equal 0 - unless the problem indicates that x cannot equal zero (e.g. x is positive).
Zero is a multiple of any integer. For example, zero is a multiple of five, as it is basically 5⋅0 - which is still 5 times an integer.
Thus, if x is a multiple of 5, then x could still equal 0 - unless the problem indicates that x cannot equal zero (e.g. x is positive).
Quotent?
Quotent is the result of division.
For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor.
For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor.
Natural Number
A natural number is simply a positive integer.
no fractions, negatives, decimals.
no fractions, negatives, decimals.
Is 1 a prime number?
1 is often mistakenly considered prime, because it is divisible by 1 and itself, but those are not two distinct factors - they're the same factor. Therefore, by definition, 1 is not prime.
What is the smallest prime number?
2
And this is the only even prime!
And this is the only even prime!
What are the 8 prime numbers under 20?
2, 3, 5, 7, 11, 13, 17, 19.
Is a # divisible by 3?
when the sum of the digits of a number is divisible by 3, the number is divisible by 3
51 is 5+1=6 (6 is divisible by 3 so then is 51)
51 is 5+1=6 (6 is divisible by 3 so then is 51)
Definition of a prime number
A prime is a natural (positive) number with EXACTLY two DISTINCT factors: 1 and itself
0 and 1 are not prime numbers
0 and 1 are not prime numbers
Difference between prime factors and factors
Integers questions concern factors or prime factors of a number
factors of a number (all factors of a number including prime) - use the Factor Chart
prime factors of a number - use the Factor Tree.
factors of a number (all factors of a number including prime) - use the Factor Chart
prime factors of a number - use the Factor Tree.
Rule of divisibility by 8
- a number is divisible by 8 if its last 3 digits form a 3-digit number that is divisible by 8.
For example, 3,720 is divisible by 8, because 720 is divisible by 8.
1,000 is divisible by 8
For example, 3,720 is divisible by 8, because 720 is divisible by 8.
1,000 is divisible by 8
Is 20±8 a multiple of 4
multiple of 4 ± multiple of 4 - MUST be a multiple of 4.
multiple of 4 ± NOT multiple of 4 - CANNOT be a multiple of 4.
multiple of 4 ± NOT multiple of 4 - CANNOT be a multiple of 4.
can NOT multiple of n ± NOT a multiple of n a multiple of n.
Multiple of n ± multiple of n = MUST be a multiple of n.
multiple of n ± NOT multiple of n = CANNOT be a multiple of n.
NOT multiple of n ± NOT a multiple of n - May or May NOT be a multiple of n
multiple of n ± NOT multiple of n = CANNOT be a multiple of n.
NOT multiple of n ± NOT a multiple of n - May or May NOT be a multiple of n
Common divisors
Two non-zero integers have a common divisor if they are both divisible by the same positive integer.
Two primes will always have a G.C.D of 1.
A G.C.D of 1 does not mean the two integers are prime - just that they have no common divisor greater than 1. Example: 8 and 9 have a G.C.D of 1.
An integer can serve as the G.C.D of itself and another integer. Example: 6 and 12 have a G.C.D of 6.
Two primes will always have a G.C.D of 1.
A G.C.D of 1 does not mean the two integers are prime - just that they have no common divisor greater than 1. Example: 8 and 9 have a G.C.D of 1.
An integer can serve as the G.C.D of itself and another integer. Example: 6 and 12 have a G.C.D of 6.
Calculating the greatest common denominator
The GCD of two or more numbers is the product of the common factors of all numbers.
If a number is a multiple of a smaller number, the GCD is thus the smaller number.
If a number is a multiple of a smaller number, the GCD is thus the smaller number.
Finding if a number is divisible by n when n> 11
An integer n is divisible by a smaller integer g if n is divisible by all of g's prime factors.
First, find the building blocks (the prime factors) of n using the factor tree.
When testing divisibility using larger building blocks that are not prime, remember to use building blocks that have no common divisor greater than 1.
First, find the building blocks (the prime factors) of n using the factor tree.
When testing divisibility using larger building blocks that are not prime, remember to use building blocks that have no common divisor greater than 1.
Is 13,512 divisible by 12?
Yes. The building blocks of 12 are 2, 2, and 3. If 13,512 is divisible by 2, 2, and 3, it is also divisible by 12.
In order to be divisible by 12, 13,512 must have two 2s as building blocks. Two 2s make a 4.
in order to test divisibility by 12, test the integer for 3 and 4.
When testing divisibility using larger building blocks that are not prime, remember to use building blocks that have no common divisor greater than 1.
In order to be divisible by 12, 13,512 must have two 2s as building blocks. Two 2s make a 4.
in order to test divisibility by 12, test the integer for 3 and 4.
When testing divisibility using larger building blocks that are not prime, remember to use building blocks that have no common divisor greater than 1.
Calculating the # of Factors of an integer
1) Test divisibility of the integer by all integers from 1 to the square root of the integer. Use the rules of divisibility to ask "Is this Integer divisible by 1? by 2? by 3?" etc.
2) Record the small factors (the numbers that the original integer IS divisible by) in the left side of the table.
3) Count the number of small factors and multiply by 2. That's the number of factors.
Find the # of factors for 140 (nearest square is 11(121)
Number of small factors: 6 [1, 2, 4, 5, 7, and 10]
2 × Number of factors of 140: 6 × 2 = 12 factors.
2) Record the small factors (the numbers that the original integer IS divisible by) in the left side of the table.
3) Count the number of small factors and multiply by 2. That's the number of factors.
Find the # of factors for 140 (nearest square is 11(121)
Number of small factors: 6 [1, 2, 4, 5, 7, and 10]
2 × Number of factors of 140: 6 × 2 = 12 factors.
Rule of divisibility by 2
if last digit is even
Rule of divisibility by 3
Sum of its digits is divisible by 3
1455 = 1+4+5+5 = 15 is divisible by 3
1455 = 1+4+5+5 = 15 is divisible by 3
Rule of divisibility by 4
Last 2 digits is divisible by 4
12,560 => 60 / 4 yes
12,560 => 60 / 4 yes
Rule of divisibility by 5
Last digit is either 5 or 0
Rule of divisibility by 6
Satisfies rule of 2 & 3
Rule of divisibility by 7
1) multiply the # of hundreds by 2
2) Add the remaining 2 digits
5,040 => 50 "hundreds" x 2 => 100
100 + 40 => 140
7 goes into 140 20 times => yes
2) Add the remaining 2 digits
5,040 => 50 "hundreds" x 2 => 100
100 + 40 => 140
7 goes into 140 20 times => yes
Rule of divisibility by 8
Last 3 digits can be divided by 8
Rule of divisibility by 9
Sum of digits divisible by 9
1,458 => 1+4+5+8 => 18 / 9 yes
1,458 => 1+4+5+8 => 18 / 9 yes
Rule of divisibility by 10
last digit is 0
Counting # of factors of a perfect square number
1) Test divisibility of the integer by all integers from 1 to the square root of the integer. Use the rules of divisibility to ask "Is this Integer divisible by 1? by 2? by 3?" etc. If you don't know the rules of divisibility yet, don't worry about it - we'll see those later on.
2) Record the small factors (the numbers that the original integer IS divisible by) in the left side of the table.
3) Count the number of small factors and multiply by 2.
4) Subtract 1 from the total - the square root cannot be counted twice. The result is the number of factors.
2) Record the small factors (the numbers that the original integer IS divisible by) in the left side of the table.
3) Count the number of small factors and multiply by 2.
4) Subtract 1 from the total - the square root cannot be counted twice. The result is the number of factors.
What is the LCM of 12, 27, and 36?
Begin by breaking down each number into its building blocks using the factor tree:
To be divisible by 12, the LCM must include a 2, 2 and 3 as its prime factors.
To be divisible by 27, the LCM must include three 3s as its prime factors.
To be divisible by 36, the LCM must include two 2s and two 3s as its prime factors.
2, 2, 3, 3, 3 as a minimum of its building clocks.
LCM - 2×2×3×3×3=4×9×3=36×3=108.
To be divisible by 12, the LCM must include a 2, 2 and 3 as its prime factors.
To be divisible by 27, the LCM must include three 3s as its prime factors.
To be divisible by 36, the LCM must include two 2s and two 3s as its prime factors.
2, 2, 3, 3, 3 as a minimum of its building clocks.
LCM - 2×2×3×3×3=4×9×3=36×3=108.
What is the sum of ALL the factors of 49?
The factors of 49 are 49, 1, 7 and 7. Add them to get the sum i.e. 49 + 1 + 7 = 57.
NOTICE: only use the square factor once - in this case the 7.
NOTICE: only use the square factor once - in this case the 7.
If a and b are integers, is a/b an integer?
(1) a is a factor of b.
(2) a is a multiple of b.
(1) a is a factor of b.
(2) a is a multiple of b.
(1) a could be equal to b, resulting in an integer. Yes. But usually "a is a factor of b" implies that 2 is smaller than b which would lead to a fraction. NO -->Maybe
(2) If a is a multiple of b, then a is also divisible by b. a/b will therefore be an integer. YES
(2) If a is a multiple of b, then a is also divisible by b. a/b will therefore be an integer. YES
If x and y are integers, and x/y is even, then is x even?
Since the product of any integer and an even number is even, x must be even.
e*e=e
e*o=e
If x & y are integers
x/y = z
y*z=x
e*e=e
e*o=e
If x & y are integers
x/y = z
y*z=x
If x to 3=5 to 5, then x=
Take the cube root of both sides:
--> (x to 3) to 1/3 = (5 to 5) to 1/3
--> x = 5 to 5/3
--> (x to 3) to 1/3 = (5 to 5) to 1/3
--> x = 5 to 5/3
What is the cube of 5 to 5
can be written as 5 to 5/3.
GCD Greatest Common Divisor Facts
Greatest Common Divisor of two integers:
Two primes will always have a G.C.D of 1.
A G.C.D of 1 does not mean the two integers are prime - just that they have no common divisor greater than 1. Example: 8 and 9 have a G.C.D of 1.
An integer can serve as the G.C.D of itself and another integer. Example: 6 and 12 have a G.C.D of 6.
Two primes will always have a G.C.D of 1.
A G.C.D of 1 does not mean the two integers are prime - just that they have no common divisor greater than 1. Example: 8 and 9 have a G.C.D of 1.
An integer can serve as the G.C.D of itself and another integer. Example: 6 and 12 have a G.C.D of 6.
|x-b| = |b-x| remember this...
|x-3| = |3-x|
any value of "b" fits. Remember this for quickly solving problems
any value of "b" fits. Remember this for quickly solving problems