Flashcards in Maths Recap Deck (103)

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1

## How to test whether a number is prime or composite

###
Before we start off, what is a prime number and a composite number? (For people who are not sure)

Quote:

A Prime Number is a positive integer that is divisible by ONLY 2 numbers (1 and itself). Whereas, A composite number is a positive integer which has divisor(s) other than the 2 numbers (1 and itself).

Ok, coming back to the point. I will name the number as n for simplicity. Following are the steps to test whether a number is a prime or composite,
1. Identify the perfect square (P.S) closest to the n.
2. Compute the square root of P.S
3. List all prime numbers upto the computed square root
4. Check if all listed prime numbers divide n equally. If not, then n is a prime. Even if atleast one of the listed prime numbers divide n, then n is a composite.
Example:
Take n as 113. To test whether 113 is a prime,
1. 100 is the closest perfect square to 113 (Remember that you take a closest perfect square that is smaller than n itself!)
2. Square root of 100 ==> 10
3. Prime numbers upto the square root (10) ==> 2,3,5,7.
4. Check whether 2,3,5,7 divides 113. None of the numbers divide 113. So, 113 is a prime.

There is one interesting cool fact to know. I remember applying this fact in actual GMAT. It's good to learn if you don't know.

Quote:

Product of any 2 numbers = Product of LCM and HCF of those 2 numbers
Product of any 2 fractions = Product of LCM and HCF of those 2 fractions

I will try to find and post a GMAT problem that uses this concept. Please feel free to post a question if you find it.

Warning: Some people may not find this approach comfortable. Some may find it comfortable. Please follow and practice only if you are comfortable with this approach. Otherwise, please ignore it.
Sometimes, we get one type of question in GMAT where we need to calculate units digit of integers raised to some power. I found a shortcut where you could save time by remembering some patterns.
How to find unit digit of powers of numbers:
Pattern 1:
Unit's place that has digits - 2/3/7/8
Then, unit's digit repeats every 4th value. Divide the power (or index) by 4.
After dividing,
If remainder is 1, unit digit of number raised to the power 1.
If remainder is 2, unit digit of number raised to the power 2.
If remainder is 3, unit digit of number raised to the power 3.
If remainder is 0, unit digit of number raised to the power 4.
Pattern 2:
Unit's place that has digits - 0/1/5/6
Then, all powers of the number have same digit as unit's place.
For e.g., 6^1 = 6, 6^2 = 36, 6^3 = 216, 6^4 = 1296
Pattern 3:
Unit's place that has digit - 4
Then,
If power is odd --> unit's digit will be '4'
If power is even --> unit's digit will be '6'
Similarly,
Unit's place that has digit - 9
Then,
If power is odd --> unit's digit will be '9
If power is even --> unit's digit will be '1'
Example:
Let's take a long number - 122 ^ 94. Find unit's digit.
Unit's place is 2. So, it repeats every 4th term of the power.
So, divide the power by 4. 94 % 4 ==> 2 (remainder).
Raise the unit digit of the base number to the power (2 - remainder). 2^2 = 4.
Thus, 4 is the unit's digit of 122^94.
I found this approach very easy and comfortable. So, see how comfortable it is for you and apply.
Real GMAT Problem: OG-12 PS #190

We are often faced to test the divisibility of some number in the exam. Following points may help you in simplifying the process,
Divisibility Tests:
To check whether a number (say n) is divisible
By 2: unit's place of n must be 0 (OR) unit's place of n must be divisible by 2.
By 3: Sum of the digits of n must be divisible by 3.
By 4: Last 2 digits (Unit's place and ten's place) of n are 0's (OR) Last 2 digits of n must be divisible by 4.
By 5: Unit's digit must be a 5 (OR) a 0.
By 6: n must be divisible by both 2 and 3 (Follow the method used for 2 and 3).
By 8: Last 3 digits (units, tens and hundredth place) of n are 0's (OR) Last 3 digits of n is divisible by 8.
By 9: Sum of the digits of n must be divisible by 9.
By 11: (Sum of the digits of n in odd places) - (Sum of the digits of n in even places) ==> Either 0 (OR) divisible by 11.
By 12: n must be divisible by both 3 and 4 (Follow the method used for 3 and 4).
By 25: Last 2 digits (units and tens place) of n are 0's (OR) Last 2 digits of n must be divisible by 25.
By 75: n must be divisible by both 3 and 25 (Follow the method used for 3 and 25).
By 125: Last 3 digits of n are 0's (OR) are divisible by 125.
Try out examples for each divisibility to grasp better.

How to find number of factors for a POSITIVE INTEGER:
There are 2 approaches to find number of factors of an integer.
Approach #1: (Factor Pairs Method)
i. Let's take a non-perfect square number such as 32. Keep picking a number (start from 1) that divides 32 until you reach a number that is smaller than the quotient.
Small Large
1 32
2 16
4 8
Stop! If you take 8, you get 4 as quotient which is smaller than the number (8).
Therefore, there are 3*2 = 6 factor pairs or number of factors of 32.
ii. Let's take a perfect square number such as 36. Keep picking a number (start from 1) that divides 36 until you reach a number that is smaller than the quotient.
Small Large
1 36
2 18
3 12
4 9
6 6
Totally, there are 5*2 = 10 factor pairs or number of factors of 36. But, (6,6) gets repeated twice. So, deduct 1 from factor pairs i.e. 10-1 = 9 factor pairs or number of factors of 36.
Approach #2: (RECOMMENDED)
If N is expresses in terms of its prime factors as a^p * b^q * c^r, where p,q,r are positive integers, then N will have (p+1) * (q+1) * (r+1) positive factors.
Example:
i. 32 = 2^5.
No. of factors = (5+1) = 6.
ii. 1452 = 2^2 * 3 * 11^2
No. of factors = (2+1) * (1+1) * (2+1) = 18.

2

## Prime numbers

###
Prime Numbers

A prime number can be divided, without a remainder, only by itself and by 1. For example, 17 can be divided only by 17 and by 1.

Some facts:

The only even prime number is 2. All other even numbers can be divided by 2.

If the sum of a number's digits is a multiple of 3, that number can be divided by 3.

No prime number greater than 5 ends in a 5. Any number greater than 5 that ends in a 5 can be divided by 5.

Zero and 1 are not considered prime numbers.

Except for 0 and 1, a number is either a prime number or a composite number. A composite number is defined as any number, greater than 1, that is not prime.

To prove whether a number is a prime number, first try dividing it by 2, and see if you get a whole number. If you do, it can't be a prime number. If you don't get a whole number, next try dividing it by prime numbers: 3, 5, 7, 11 (9 is divisible by 3) and so on, always dividing by a prime number (see table below).

3

##
The Sieve of Eratosthenes

###
The Sieve of Eratosthenes

Eratosthenes (275-194 B.C., Greece) devised a 'sieve' to discover prime numbers. A sieve is like a strainer that you use to drain spaghetti when it is done cooking. The water drains out, leaving your spaghetti behind. Eratosthenes's sieve drains out composite numbers and leaves prime numbers behind.

To use the sieve of Eratosthenes to find the prime numbers up to 100, make a chart of the first one hundred positive integers (1-100):

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Cross out 1, because it is not prime.

Circle 2, because it is the smallest positive even prime. Now cross out every multiple of 2; in other words, cross out every second number.

Circle 3, the next prime. Then cross out all of the multiples of 3; in other words, every third number. Some, like 6, may have already been crossed out because they are multiples of 2.

Circle the next open number, 5. Now cross out all of the multiples of 5, or every 5th number.

Continue doing this until all the numbers through 100 have either been circled or crossed out. You have just circled all the prime numbers from 1 to 100!

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## How to test whether a number is prime or composite

###
1. Identify the perfect square (P.S) closest to the n.

2. Compute the square root of P.S

3. List all prime numbers upto the computed square root

4. Check if all listed prime numbers divide n equally. If not, then n is a prime. Even if atleast one of the listed prime numbers divide n, then n is a composite.

Example:
Take n as 113. To test whether 113 is a prime,

1. 100 is the closest perfect square to 113 (Remember that you take a closest perfect square that is smaller than n itself!)

2. Square root of 100 ==> 10

3. Prime numbers upto the square root (10) ==> 2,3,5,7.

4. Check whether 2,3,5,7 divides 113. None of the numbers divide 113. So, 113 is a prime.

5

## How to calculate LCM and HCF of fractions:

###

Quote:

L.C.M of 2 fractions = L.C.M of NUMERATORS / H.C.F of DENOMINATORS

H.C.F of 2 fractions = H.C.F of NUMERATORS / L.C.M of DENOMINATORS

6

##
Product of two numbers

Product of two fractions

###
Quote:

Product of any 2 numbers = Product of LCM and HCF of those 2 numbers

Product of any 2 fractions = Product of LCM and HCF of those 2 fractions

I will try to find and post a GMAT problem that uses this concept. Please feel free to post a question if you find it.

7

## How to find unit digit of powers of numbers

###
Pattern 1:

Unit's place that has digits - 2/3/7/8

Then, unit's digit repeats every 4th value. Divide the power (or index) by 4.

After dividing,

If remainder is 1, unit digit of number raised to the power 1.

If remainder is 2, unit digit of number raised to the power 2.

If remainder is 3, unit digit of number raised to the power 3.

If remainder is 0, unit digit of number raised to the power 4.

Pattern 2:

Unit's place that has digits - 0/1/5/6

Then, all powers of the number have same digit as unit's place.

For e.g., 6^1 = 6, 6^2 = 36, 6^3 = 216, 6^4 = 1296

Pattern 3:

Unit's place that has digit - 4

Then,

If power is odd --> unit's digit will be '4'

If power is even --> unit's digit will be '6'

Similarly,

Unit's place that has digit - 9

Then,

If power is odd --> unit's digit will be '9

If power is even --> unit's digit will be '1'

Example:

Let's take a long number - 122 ^ 94. Find unit's digit.

Unit's place is 2. So, it repeats every 4th term of the power.

So, divide the power by 4. 94 % 4 ==> 2 (remainder).

Raise the unit digit of the base number to the power (2 - remainder). 2^2 = 4.

Thus, 4 is the unit's digit of 122^94.

I found this approach very easy and comfortable. So, see how comfortable it is for you and apply.

8

##
Area of triangle

Area of rectangle

Area of trapezoid

Area of elipse

###
Triangle

Area = ½ × b × h

b = base

h = vertical height

Rectangle

Area = w × h

w = width

h = height

Trapezoid (US)

Trapezium (UK)

Area = ½(a+b) × h

h = vertical height

Ellipse

Area = πab

9

##
Area of square

Area of parallelogram

Area of circle

Area of sector

###
Square

Area = a2

a = length of side

Parallelogram

Area = b × h

b = base

h = vertical

Circle

Area = π × r2

Circumference = 2 × π × r

r = radius

Sector

Area = ½ × r2 × θ

r = radius

θ = angle in radians

10

##
Recognize multiples of

2

3

4

5

6

9

###
2. Last digit is even

3. Sum of digits is a multiple of 3

4. Last two digits are multiples of 4

5. Last digit is 5 or 0

6. Sum of digits is a multiple of 3 and the last digit is even

9. Sum of digits is a multiple of 9

10. Last digit is 0

12. Sum of digits is a multiple of 3 and the last two digits are a multiple of 4

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##
Isosceles triangle

Equilateral triangle

Pythagorean theorem

30-60-90 triangle

45-45-90 triangle

###
Isosceles triangle - two equal sides and two equal angles

Equilateral triangle- all sides equal and all equal angles

Pythagorean theorem- a^2 + b^2 = c^2

30-60-90 triangle - 1 / root 3 / 2

45-45-90 triangle - 1 / 1 / root 2

12

##
Slope

Permutation

Combination

###
Slope = change in y / change in x

Permutation - n! / (n-k)!

Combination- n!/[k!(n-k)!]

13

##
Sum of all angles of a regular polygon

Area of sector

Volume of cylinder

Volume of sphere

###
Sum of angles = (n-2)*180

area of Sector - r/360 * Pi * r ^2

Volume of cylinder - Pi * r^2 * h

Volume of sphere- 4/3 * Pi * r^3

14

## Squares of 2 till 10

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2-4

3-9

4-32

5-25

6-36

7-49

8-64

9-81

10-100

15

##
Squares of 11 to 15

###
11-121

12-144

13-169

14-212

15-225

16

## Squares of 16 to 20

###
16-256

17-289

18-324

19-361

20-400

17

## Squares of 21 to 25

###
21-441

22-484

23-529

24-592

25-625

18

## Cubes of 1 to 5

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1-1

2-8

3-27

4-64

5-125

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## Cubes of 6 to 10

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6-216

7-343

8-512

9-729

10-1000

20

## Cubes of 11 to 15

###
11-1331

12-1728

13-2197

14-2744

15-3375

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## Divisibility Tests:

###
To check whether a number (say n) is divisible

By 2: unit's place of n must be 0 (OR) unit's place of n must be divisible by 2.

By 3: Sum of the digits of n must be divisible by 3.

By 4: Last 2 digits (Unit's place and ten's place) of n are 0's (OR) Last 2 digits of n must be divisible by 4.

By 5: Unit's digit must be a 5 (OR) a 0.

By 6: n must be divisible by both 2 and 3 (Follow the method used for 2 and 3).

By 8: Last 3 digits (units, tens and hundredth place) of n are 0's (OR) Last 3 digits of n is divisible by 8.

By 9: Sum of the digits of n must be divisible by 9.

By 11: (Sum of the digits of n in odd places) - (Sum of the digits of n in even places) ==> Either 0 (OR) divisible by 11.

By 12: n must be divisible by both 3 and 4 (Follow the method used for 3 and 4).

By 25: Last 2 digits (units and tens place) of n are 0's (OR) Last 2 digits of n must be divisible by 25.

By 75: n must be divisible by both 3 and 25 (Follow the method used for 3 and 25).

By 125: Last 3 digits of n are 0's (OR) are divisible by 125.

Try out examples for each divisibility to grasp better.

22

## How to find number of factors for a POSITIVE INTEGER:

###

There are 2 approaches to find number of factors of an integer.

Approach #1: (Factor Pairs Method)

i. Let's take a non-perfect square number such as 32. Keep picking a number (start from 1) that divides 32 until you reach a number that is smaller than the quotient.

Small Large

1 32

2 16

4 8

Stop! If you take 8, you get 4 as quotient which is smaller than the number (8).

Therefore, there are 3*2 = 6 factor pairs or number of factors of 32.

ii. Let's take a perfect square number such as 36. Keep picking a number (start from 1) that divides 36 until you reach a number that is smaller than the quotient.

Small Large

1 36

2 18

3 12

4 9

6 6

Totally, there are 5*2 = 10 factor pairs or number of factors of 36. But, (6,6) gets repeated twice. So, deduct 1 from factor pairs i.e. 10-1 = 9 factor pairs or number of factors of 36.

Approach #2: (RECOMMENDED)

If N is expresses in terms of its prime factors as a^p * b^q * c^r, where p,q,r are positive integers, then N will have (p+1) * (q+1) * (r+1) positive factors.

Example:

i. 32 = 2^5.

No. of factors = (5+1) = 6.

ii. 1452 = 2^2 * 3 * 11^2

No. of factors = (2+1) * (1+1) * (2+1) = 18.

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## Number of factors

###
If N is a perfect square, then the number of factors of N will ALWAYS be an ODD number.

If N is a NON-perfect square, then the number of factors of N will ALWAYS be an EVEN number.

_________________

Download GMAT Math and CR questions with Solutions from Instructors and High-scorers:

http://www.beatthegmat.com/download-gmat-questions-with-expert-solutions-t59366.html

-----------

GO GREEN..! GO VEG..!

Daily Quote:

“Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life”

24

## How to find Sum of all factors of a POSITIVE integer

###
How to find Sum of all factors of a POSITIVE integer:

If N is expressed in terms of its prime factors as a^p * b^q * c^r, where p,q,r are positive integers, then the sum of all factors of N is

[ (a^(p+1) - 1) / a-1 ] * [ (b^(q+1) - 1) / b-1 ] * [ (c^(r+1) - 1) / c-1 ]

25

## Factorization rule

###
Any number whose prime factorization contains even powers of primes, then the number must be a perfect square.

Any number whose prime factorization contains powers of primes with multiples of 3, then the number must be a perfect cube

26

## Remainders

###
Guys are indeed following the SC thread. I hope people are also following this thread. Let me continue to post flashcards.

REMAINDERS:

(I)

When 2 numbers are divided by same divisor and the remainders obtained are the same,

THEN

DIFFERENCE b/w 2 numbers is also divisible by that divisor.

(II)

When 2 positive numbers 'a' and 'b' are divided by the same divisor 'd' and remainders obtained are 'r1' and 'r2' respectively,

THEN

the remainders obtained when a+b is divided by d will be r1+r2

Quote:

NOTE: If r1+r2 >= d, compute (r1+r2) - d as the remainder.

(III)

When 2 positive numbers 'a' and 'b' are divided by the same divisor 'd' and the remainders obtained are 'r1' and 'r2' respectively,

THEN

the remainders obtained when a*b is divided by d will be r1*r2

Quote:

NOTE: If r1*r2 >= d, compute (r1*r2) / d as the remainder.

TAKEAWAY:

A remainder can NEVER be greater than or equal to the divisor.

27

## How to find REMAINDER for LARGE POWERS of numbers:

###

There are 2 ways to do so:

1. Pattern Method:

Example:

What is the remainder when 2^56 / 7 ?

Solution:

Remainder when 2^1 is divided by 7 is 2

Remainder when 2^2 is divided by 7 is 4

Remainder when 2^3 is divided by 7 is 1

Remainder when 2^4 is divided by 7 is 2 --> Repeats again.

The remainder repeats after 3 steps i.e. in the 4th step.

Now, Divide the power (or index) by 3 (no of steps after which remainder repeats) and compute a new remainder.

56 % 3 --> 2 (remainder)

Now, raise the base (2) to the power 2 (new remainder). 2^2 % 7 --> 4.

Thus, 4 is the remainder when 2^56 / 7.

2. Remainder Theorem Method: (NOT RECOMMENDED unless clear)

Example:

What is the remainder when 2^51 / 7 ?

Solution:

2^51 can be changed to (2^3)^17.

7 can be changed to (8-1) OR (2^3 - 1)

Substitute 'x' in place of 2^3,

x^17 / (x-1)

Remainder is f(1). Substitute 1 in 'x',

Remainder is 1.

Thus, 1 is the remainder when 2^51 / 7.

28

## Simple Facts:

###
Simple Facts:

a^n - b^n:

1. ALWAYS divisible by a-b

2. If n is even, it is divisible by a+b

3. If n is odd, it is NOT divisible by a+b

a^n + b^n:

1. NEVER divisible by a-b

2. If n is odd, it is divisible by a+b

3. If n is even, it is NOT divisible by a+b

29

## Multiples of N

###
Playing with Multiples of N:

(I)

If you add/subtract multiples of number 'N', the result is also a multiple of 'N'.

Examples:

35+21 = 56 [Multiple of 7]

20-15 = 5 [Multiple of 5]

TAKEAWAY:

In general, if N is a divisor of both x and y, then N is a divisor of both x+y and x-y.

(II)

If you add/subtract a multiple of N to/from a non-multiple of N, the result is a non-multiple of N.

Example:

9-5 = 4 [(Multiple of 3) - (Non-Multiple of 3) = (Non-multiple of 3)]

(III)

If you add/subtract 2 non-multiples of N, the result could either be a multiple or a non-multiple of N.

Examples:

19+13 = 32 [(Non-Multiple of 3) - (Non-Multiple of 3) = (Non-multiple of 3)]

19+14 = 33 [(Non-Multiple of 3) - (Non-Multiple of 3) = (Multiple of 3)]

EXCEPTION:

When N = 2, two odds always sum to an even number (Multiple of 2).

30