Gmat - Number Properties - Even, Odds, Positives, Negatives, Arithmetic, Remainders

Front 1

What are the 3 main Arithmetical Rules?

Back 1

• The sum/difference/product of 2 integers is always an integer
  
• The quotient of two integers is not always an integer
  
• An integer is said to be divisible by another if the result is also an integer
  

Front 2

multiply a # by 5 is the same as...

Back 2

# x 10 / 2

Front 3

multiply a # by 25 is the same as...

Back 3

# x 100 / 4

Front 4

multiply a # by 45 is the same as...

Back 4

# x 90/ 2

Front 5

multiply a # by 75 is the same as...

Back 5

# x 300 / 4

Front 6

multiply a # by 225 is the same as...

Back 6

# x 900 / 4

Front 7

multiply a # by 275 is the same as...

Back 7

# x 1100 / 4

Front 8

multiply a # by 125 is the same as...

Back 8

# x 500 / 4

Front 9

multiply a # by 22 is the same as...

Back 9

# x 11 * 2

Front 10

multiply a # by 33 is the same as...

Back 10

# x 11 * 3

Front 11

777 x 11 = quick technique

Back 11

work it backwards

777 x 11 = (rest of first addition + 2nd term + 3rd term) & (1st term + 2nd term) & 1st 7

7 = 7

7 + 7 = "1"4

7 + 7 + 1 = "1"5

7 + 1 = 8

8547

Front 12

8²

Back 12

64

Front 13

9²

Back 13

81

Front 14

11²

Back 14

121

Front 15

12²

Back 15

144

Front 16

13²

Back 16

169

Front 17

14²

Back 17

196

Front 18

15²

Back 18

225

Front 19

16²

Back 19

256

Front 20

17²

Back 20

289

Front 21

18²

Back 21

324

Front 22

19²

Back 22

361

Front 23

21²

Back 23

441

Front 24

22²

Back 24

484

Front 25

23²

Back 25

529

Front 26

24²

Back 26

576

Front 27

25²

Back 27

625

Front 28

Fraction, Decimal and Percentage

1 / 2

Back 28

0.5 --- 50%

Front 29

Fraction, Decimal and Percentage

1/3

Back 29

0.33 --- 33 1/3%

Front 30

Fraction, Decimal and Percentage

2/3

Back 30

0.66 --- 66 2/3%

Front 31

Fraction, Decimal and Percentage

1/4

Back 31

0.25 --- 25%

Front 32

Fraction, Decimal and Percentage

3/4

Back 32

0.75 --- 75%

Front 33

Fraction, Decimal and Percentage

1/5

Back 33

0.2 --- 20%

Front 34

Fraction, Decimal and Percentage

2/5

Back 34

0.4 --- 40%

Front 35

Fraction, Decimal and Percentage

3/5

Back 35

0.6 --- 60%

Front 36

Fraction, Decimal and Percentage

4/5

Back 36

0.8 --- 80%

Front 37

Fraction, Decimal and Percentage

1/6

Back 37

0.1666 --- 16 2/3%

Front 38

Fraction, Decimal and Percentage

5/6

Back 38

0.83 --- 83 2/3%

Front 39

Fraction, Decimal and Percentage

1/8

Back 39

0.125 --- 12 1/2

Front 40

Fraction, Decimal and Percentage

3/8

Back 40

0.375 --- 37 1/2%

Front 41

Fraction, Decimal and Percentage

5/8

Back 41

0.625 --- 62 1/2%

Front 42

Fraction, Decimal and Percentage

7/8

Back 42

0.875 --- 87 1/2%

Front 43

Fraction, Decimal and Percentage

1/9

Back 43

0.1111 --- 11 1/9%

Front 44

√2

Back 44

1.4

Front 45

√3

Back 45

1.7

Front 46

PI

Back 46

22/7 OR 3.14

Front 47

Divisibility and Primes - An integer is divisible by:

2

Back 47

if it is EVEN

Front 48

Divisibility and Primes - An integer is divisible by:

3

Back 48

if the sum of integer’s digits is divisible by 3

Front 49

Divisibility and Primes - An integer is divisible by:

4

Back 49

if the integer is div. by 2 twice or if THE LAST TWO DIGITS are divisible by 4

Front 50

Divisibility and Primes - An integer is divisible by:

6

Back 50

if the integer is divisible BOTH by 2 and 3

Front 51

Divisibility and Primes - An integer is divisible by:

8

Back 51

if the integer is divisible by 2 three times or IF THE LAST 3 DIGITS are divisible by 8

Front 52

Divisibility and Primes - An integer is divisible by:

9

Back 52

if the sum of the integer digits is divisible by 9

Front 53

Divisibility and Primes - An integer is divisible by:

11

Back 53

sum of digits in odd places – sum of digits in even places either = 0 or divisible by 11.

Front 54

Addition/subtraction - Odds and Evens

Addition/subtraction: If you add/subtract 2 odds or 2 even, the result will be...

Back 54

even

Front 55

Addition/subtraction - Odds and Evens

If you add/subtract an odd with an even, result will be ...

Back 55

odd

Front 56

Multiplication/division - Positives and Negatives

When you multiply or divide a group of nonzero numbers, the result will be positive if you have...

Back 56

...an even number of negative numbers. If not, the result will be negative.

Front 57

Multiplication/division - Positives and Negatives

Is the product of all elements in set S negative?

(1) All the elements in set S are negative

(2) There are five negative elements in S.

Back 57

This is a trap, as statement 2 seems sufficient, but it is not as we don’t know whether there is a 0 on set S or not. Both statements together are sufficient.

Front 58

Multiplication/division - Odds and Evens

If any of the integers in a multiplication is even, result is ...

Back 58

even

Front 59

Multiplication/division - Odds and Evens

An even divided by another even can yield an

Back 59

even, odd or non-integer

Front 60

Multiplication/division -Division of Odds and Evens

An odd divided by an even will always yield a

Back 60

non-integer

Front 61

Multiplication/division - Division of Odds and Evens

An even divided by an odd cannot yield an ...

Back 61

Odd number

(either even result or non-integer result)ie

Front 62

Multiplication/division - Division of Odds and Evens

An odd divided by another odd cannot yield an

Back 62

even number

Front 63

Divisibility and Primes - Prime numbers

Since 2 is the only even prime number, the sum/difference of two primes can only result an......... if one of them is 2.

Back 63

odd number

For all other cases, the product will be odd and the sum/difference will be even.

Front 64

Addition/subtraction/multiplication - Positives and Negatives

Is the product of all elements in set S negative?

(1) All the elements in set S are negative

(2) There are five negative elements in S.

Back 64

This is a trap, as statement 2 seems sufficient, but it is not as we don’t know whether there is a 0 on set S or not. Both statements together are sufficient.

Front 65

Addition/subtraction/multiplication - Positives and Negatives

If ab > 0, which of the following is negative?

(a) a +b (b) |a| + b (c) b –a (d)a/b (e)–a/b

Back 65

To solve these problems, build a positive/negative table and if necessary, pick numbers

Picking 2 sets of numbers was sufficient to solve. Answer: E

Front 66

Addition/subtraction/multiplication - Odds and Evens

Example: If a, b and c are integers and a.b + c is odd, what is true?

I. a + c is odd

II. b + c is odd

III. a.b.c is even

Back 66

Therefore, a + c can be odd or even, and the same happens for b + c. So, the only necessarily true statement is III.

Front 67

Divisibility - general formula

Divisibility most important formula

Back 67

Pure Gold

Front 68

Divisibility - remainder techniques

The remainder of any number ALWAYS MUST be ...and ... than the divisor

Back 68

non-negative and smaller than the divisor.

Front 69

Divisibility and Primes - Advanced remainder techniques

Which operations can I to the remainder of any number?

Back 69

You can add, subtract and multiply remainder,

as long as you correct the final result by dividing the resultant number by the divisor.

Front 70

Divisibility and Primes - Advanced remainder techniques

If you need a number that leaves a remainder of R when divided by N.

Back 70

simply take any multiple of N and add R

Front 71

Divisibility and Primes - Remainders

On simpler remainder problems, the best technique is...

Back 71

...to pick numbers.

For example, consider that the Remainder is always less than the denominator and the sum of denominator and remainder is a multiple of the numerator

Front 72

How many #'s up to 100 are divisible by 6?

Back 72

Front 73

Back 73

Front 74

Divisibility and Primes - Advanced remainder techniques

If A/B yields 4.35, what could be the remainder?

(A) 13 (B) 14 (C) 15 (D) 16 (E) 17

Back 74

First, convert .35 to a fraction: .35 = 35/100 = 7/20. Now, compare this fraction with B (the divisor on this problem):

7/20 = R/B

From this equation, we know that the remainder must be a multiple of 7. The correct answer is 14.

Front 75

Divisibility and Primes - Advanced remainder techniques

What is the formula that equals the decimal part of the quotient to remainder?

Back 75

What’s crucially important is — the decimal part of the decimal quotient equals the final fraction:

Front 76

Advanced Problems - Representing Evens and Odds algebraically

What is the remainder of a/4?

(1) a is the square of an odd integer

(2) a is a multiple of 3

Back 76

if we square (2n+1) we have 4n² + 4n + 1, so it is clear that any odd number, when squared and further divided by 4 will leave a remainder of 1.

So, (1) is SUFFICIENT. Statement (2) is INSUFFICIENT, as a could be an even number, like 6 or 12.

Front 77

Divisibility and addition/subtraction

If you add or subtract multiples of an integer, the result is...

Back 77

also a multiple

Front 78

Advanced Problems - Divisibility and addition/subtraction

When you add two non-multiples of 2, the result is ...

Back 78

is a multiple of 2

Front 79

Divisibility and addition/subtraction

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

Back 79

non-multiple of N

Front 80

Advanced Problems - Odd/Even, Pos/neg, Consec Integers

If x, y and z are integers and xyz is divisible by 8, is x even?
(1) xy is divisible by 4
(2) x, y and z are all not divisible by 4

Back 80

(1) is not sufficient, as y could be divisible by 4 and x could be odd.

(2) is SUFFICIENT, as if neither x, y and z are divisible by 4, all of them must be divisible by 2 in order to let xyz be divisible by 8. Answer: B

Front 81

Divisibility and Primes - Prime numbers

Conversely, knowing that the product of two prime numbers is even OR that the sum/difference is odd is...

Back 81

sufficient to know that one of the prime numbers is 2.

Front 82

Divisibility and Primes - Multiplication of Odds and Evens

If many even integers are multiplied, the result is...

Back 82

higher and higher powers of 2

This is because each integer will have 2 as a factor

Front 83

Divisibility and Primes - Prime

If x>1, what is x?

(1) There are x unique factors of x

(2) x plus any prime larger than x is odd.

Back 83

Statement (1) is sufficient as this property holds only for 1 and 2. Statement (2) only tells us that x is even, as it is not stated in the problem that x is prime.

(1) x can be divided from 1, 2, 3,... x- 1, and x, so the only two possible numbers for that to happen is by plugging x=2 to x / x - 1 which yields 2 and 1, but the constraints says x>1. So the ans is 2. If x=3.then no solution x=4 no solution either