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GMAT Number Properties

Subjects : gmat, math

Instructions:

Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. How to solve: For any positive integer n - the sum of the 1st n positive integers equals n(n+1)/2. What is the sum of all the even integers between 99 and 301? (A) 10 -100 (B) 20 -200 (C) 22 -650 (D) 40 -200 (E) 45 -150
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹
1. The smallest or largest element 2. The increment 3. The number of items in the set
The sum of EVEN INTEGERS between 99 and 301 is the sum of EVEN INTEGERS between 100 and 300 - or the sum of the 50th EVEN INTEGER through the 150th EVEN INTEGER.To get this sum: -Find the sum of the FIRST 150 even integers (ie 2 times the sum of the
ODD

2. The sum of any two primes will be ____ - unless ______.
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.
31 -37
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.
The sum of any two primes will be even - unless one of the two primes is 2.

3. Prime Numbers:3x
11 -13 -17 -19
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6
[(last - first) / increment] + 1
31 -37

4. How to solve: If k - m - and t are positive integers and k/6 + m/4 = t/12 - do t and 12 have a common factor greater than 1? 1. k is a multiple of 3 2. m is a multiple of 3

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5. The PRODUCT of n consecutive integers is divisible by ____.
NEVER CONTRADICT ONE ANOTHER
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6
The PRODUCT of n consecutive integers is divisible by n!.
The same sign as the base

6. v2˜
1.4
Prime
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.
The PRODUCT of n consecutive integers is divisible by n!.

7. In an evenly spaced set - the average can be found by finding ________.
The middle number
83 -89
Prime factorization
EVEN

8. Let N be an integer. If you add two non-multiples of N - the result could be _______.
A PERFECT SQUARE
The PRODUCT of n consecutive integers is divisible by n!.
Never prime
Either a multiple of N or a non-multiple of N

9. The two statements in a data sufficiency problem will _______________.
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.
In an evenly spaced set - the average and the median are equal.
41 -43 -47
NEVER CONTRADICT ONE ANOTHER

10. Positive integers with only two factors must be ___.
The same sign as the base
15
71 -73 -79
Prime

11. v169=
Prime factorization
13
The sum of any two primes will be even - unless one of the two primes is 2.
Put the coefficient under the radical to get a better approximation

12. The average of an ODD number of consecutive integers will ________ be an integer.
Never prime
83 -89
The average of an ODD number of consecutive integers will ALWAYS be an integer.
53 -59

13. v5˜
Look at the numbers from 1 to 30 - inclusive - that have at least one factor of 3 and count up how many each has: 3-1; 6-1; 9-2; 12-1; 15-1; 18-2; 21-1; 24-1; 27-3; 30-1 - The answer is 14.
Prime
The average of an EVEN number of consecutive integers will NEVER be an integer.
2.5

14. Prime factors of _____ must come in pairs of three.
PERFECT CUBES
A non-multiple of N.
1.4
1.7

15. If the problem states/assumes that a number is an integer - check to see if you can use _______.
N is a divisor of x+y
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹
The sum of EVEN INTEGERS between 99 and 301 is the sum of EVEN INTEGERS between 100 and 300 - or the sum of the 50th EVEN INTEGER through the 150th EVEN INTEGER.To get this sum: -Find the sum of the FIRST 150 even integers (ie 2 times the sum of the
Prime factorization

16. Prime Numbers:2x
Either a multiple of N or a non-multiple of N
53 -59
61 -67
23 -29

17. v196=
A PERFECT SQUARE
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.
N is a divisor of x+y
14

18. Prime Numbers:6x
83 -89
16
ONLY the nonnegative root of the numberUNLIKE
61 -67

19. If estimating a root with a coefficient - _____ .
15
[(last - first) / increment] + 1
Put the coefficient under the radical to get a better approximation
EVEN

20. v256=
23 -29
16
Put the coefficient under the radical to get a better approximation
The middle number

21. The prime factorization of a perfect square contains only ______ powers of primes.
11 -13 -17 -19
A PERFECT SQUARE
EVEN
3·3n = 3^{n+1}

22. Prime Numbers:7x
1. The smallest or largest element 2. The increment 3. The number of items in the set
FACTOR
A PERFECT SQUARE
71 -73 -79

23. The formula for finding the number of consecutive multiples in a set is _______.
EVEN
2 -3 -5 -7
[(last - first) / increment] + 1
N is a divisor of x+y

24. If N is a divisor of x and y - then _______.
N is a divisor of x+y
2 -3 -5 -7
A PERFECT SQUARE
97

25. 3n + 3n + 3n = _____ = ______
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.
3·3n = 3^{n+1}
41 -43 -47
31 -37

26. In an evenly spaced set - the ____ and the ____ are equal.
[(last - first) / increment] + 1
A PERFECT SQUARE
In an evenly spaced set - the average and the median are equal.
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.

27. In an evenly spaced set - the mean and median are equal to the _____ of _________.
PERFECT CUBES
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.
11 -13 -17 -19
The average of an ODD number of consecutive integers will ALWAYS be an integer.

28. How to find the sum of consecutive integers:
EVEN
1. Average the first and last to find the mean. 2. Count the number of terms. 3. Multiply the mean by the number of terms.
A PERFECT SQUARE
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6

29. How to solve: Is the integer z divisible by 6? (1) gcd(z -12) = 3 (2) gcd(z -15) = 15
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6
A non-multiple of N.
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹
PERFECT CUBES

30. N! is _____ of all integers from 1 to N.
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.
A MULTIPLE
FACTOR
The average of an ODD number of consecutive integers will ALWAYS be an integer.

31. For ODD ROOTS - the root has ______.
25
In an evenly spaced set - the average and the median are equal.
A MULTIPLE
The same sign as the base

32. The average of an EVEN number of consecutive integers will ________ be an integer.
The average of an EVEN number of consecutive integers will NEVER be an integer.
The middle number
If 2 cannot be one of the primes in the sum - the sum must be even.
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.

33. If 2 cannot be one of the primes in the sum - the sum must be _____.
71 -73 -79
Express as 2k + 3m = t. 1. If k is a multiple of 3 - then so is t and we have a yes. => S 2. If m is a multiple of 3 - we don't know. => I A/1 Alone.
A non-multiple of N.
If 2 cannot be one of the primes in the sum - the sum must be even.

34. Any integer with an EVEN number of total factors cannot be ______.
11 -13 -17 -19
A PERFECT SQUARE
1.4
Express as 2k + 3m = t. 1. If k is a multiple of 3 - then so is t and we have a yes. => S 2. If m is a multiple of 3 - we don't know. => I A/1 Alone.

35. How to test for sufficiency: If p is an integer - is p/n an integer? (1) k1p/n is an integer(2) k2p/n is an integer
ODD
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.
31 -37
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.

36. v225=
The sum of EVEN INTEGERS between 99 and 301 is the sum of EVEN INTEGERS between 100 and 300 - or the sum of the 50th EVEN INTEGER through the 150th EVEN INTEGER.To get this sum: -Find the sum of the FIRST 150 even integers (ie 2 times the sum of the
14
15
31 -37

37. All evenly spaced sets are fully defined if:1. _____ 2. _____ 3. _____ are known.
1. The smallest or largest element 2. The increment 3. The number of items in the set
2 -3 -5 -7
1.4
The same sign as the base

38. Prime Numbers:1x
The PRODUCT of n consecutive integers is divisible by n!.
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.
11 -13 -17 -19
83 -89

39. How to solve: If p is the product of the integers from 1 to 30 - inclusive - what is the greatest integer n for which 3n is a factor of p?
Look at the numbers from 1 to 30 - inclusive - that have at least one factor of 3 and count up how many each has: 3-1; 6-1; 9-2; 12-1; 15-1; 18-2; 21-1; 24-1; 27-3; 30-1 - The answer is 14.
EVEN
14
FACTOR

40. v625=
The average of the set times the number of elements in the set
25
14
11 -13 -17 -19

41. In an evenly spaced set - the sum of the terms is equal to ____.
31 -37
FACTOR
1.7
The average of the set times the number of elements in the set

42. Let N be an integer. If you add a multiple of N to a non-multiple of N - the result is ________.
1.7
13
A non-multiple of N.
Put the coefficient under the radical to get a better approximation

43. Prime Numbers:8x
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.
NEVER CONTRADICT ONE ANOTHER
1.4
83 -89

44. ³v216 =
ONLY the nonnegative root of the numberUNLIKE
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.
13

45. On data sufficiency - ALWAYS _______ algebraic expressions when you can. ESPECIALLY for divisibility.
PERFECT CUBES
A MULTIPLE
2 -3 -5 -7
FACTOR

46. Any integer with an ODD number of total factors must be _______.
A PERFECT SQUARE
A non-multiple of N.
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.
1.4

47. The prime factorization of __________ contains only EVEN powers of primes.
41 -43 -47
The sum of any two primes will be even - unless one of the two primes is 2.
The middle number
A PERFECT SQUARE

48. All perfect squares have a(n) _________ number of total factors.
[(last - first) / increment] + 1
53 -59
Never prime
ODD

49. Positive integers with more than two factors are ____.
The average of an EVEN number of consecutive integers will NEVER be an integer.
ONLY the nonnegative root of the numberUNLIKE
1. Average the first and last to find the mean. 2. Count the number of terms. 3. Multiply the mean by the number of terms.
Never prime

50. Prime Numbers:4x
13
41 -43 -47
61 -67
1.7

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GMAT Number Properties

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