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GMAT Number Properties

Subjects : gmat, math

Instructions:

Answer 50 questions in 15 minutes.
If you are not ready to take this test, you can study here.
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. Prime Numbers:9x
PERFECT CUBES
61 -67
97
Prime

2. v256=
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹
2 -3 -5 -7
EVEN
16

3. Any integer with an ODD number of total factors must be _______.
A PERFECT SQUARE
Prime
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.
83 -89

4. v196=
A MULTIPLE
13
14
Prime factorization

5. Prime Numbers:4x
PERFECT CUBES
41 -43 -47
14
11 -13 -17 -19

6. If N is a divisor of x and y - then _______.
The average of an EVEN number of consecutive integers will NEVER be an integer.
15
In an evenly spaced set - the average and the median are equal.
N is a divisor of x+y

7. In an evenly spaced set - the average can be found by finding ________.
41 -43 -47
The middle number
2.5
NEVER CONTRADICT ONE ANOTHER

8. v2˜
1.4
11 -13 -17 -19
If 2 cannot be one of the primes in the sum - the sum must be even.
2.5

9. 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
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.
25
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.
11 -13 -17 -19

10. 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
1.4
Either a multiple of N or a non-multiple of N
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
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.

11. 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

12. All perfect squares have a(n) _________ number of total factors.
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹
14
ODD
The middle number

13. Prime Numbers:5x
A MULTIPLE
Prime
53 -59
11 -13 -17 -19

14. In an evenly spaced set - the mean and median are equal to the _____ of _________.
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.
A PERFECT SQUARE
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.
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.

15. If 2 cannot be one of the primes in the sum - the sum must be _____.
If 2 cannot be one of the primes in the sum - the sum must be even.
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹
The middle number
16

16. For ODD ROOTS - the root has ______.
13
The same sign as the base
In an evenly spaced set - the average and the median are equal.
16

17. Prime Numbers:0x
Prime
2 -3 -5 -7
The PRODUCT of n consecutive integers is divisible by n!.
1.4

18. Prime Numbers:2x
1. The smallest or largest element 2. The increment 3. The number of items in the set
23 -29
The PRODUCT of n consecutive integers is divisible by n!.
A MULTIPLE

19. Let N be an integer. If you add two non-multiples of N - the result could be _______.
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.
In an evenly spaced set - the average and the median are equal.
Either a multiple of N or a non-multiple of N
1.4

20. N! is _____ of all integers from 1 to N.
The PRODUCT of n consecutive integers is divisible by n!.
PERFECT CUBES
A MULTIPLE
Put the coefficient under the radical to get a better approximation

21. v5˜
25
2.5
ODD
1.7

22. All evenly spaced sets are fully defined if:1. _____ 2. _____ 3. _____ are known.
2.5
1. The smallest or largest element 2. The increment 3. The number of items in the set
The average of the set times the number of elements in the set
FACTOR

23. Prime Numbers:3x
PERFECT CUBES
31 -37
3·3n = 3^{n+1}
1. The smallest or largest element 2. The increment 3. The number of items in the set

24. The prime factorization of __________ contains only EVEN powers of primes.
In an evenly spaced set - the average and the median are equal.
The same sign as the base
41 -43 -47
A PERFECT SQUARE

25. In an evenly spaced set - the ____ and the ____ are equal.
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.
Either a multiple of N or a non-multiple of N
1.7
In an evenly spaced set - the average and the median are equal.

26. The prime factorization of a perfect square contains only ______ powers of primes.
71 -73 -79
EVEN
1. The smallest or largest element 2. The increment 3. The number of items in the set
31 -37

27. When we take an EVEN ROOT - a radical sign means ________. This is _____ even exponents.
In an evenly spaced set - the average and the median are equal.
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹
ONLY the nonnegative root of the numberUNLIKE
97

28. The average of an ODD number of consecutive integers will ________ be an integer.
The average of an EVEN number of consecutive integers will NEVER be an integer.
53 -59
The average of an ODD number of consecutive integers will ALWAYS be an integer.
41 -43 -47

29. v625=
25
ONLY the nonnegative root of the numberUNLIKE
Prime factorization
71 -73 -79

30. The formula for finding the number of consecutive multiples in a set is _______.
[(last - first) / increment] + 1
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 PERFECT SQUARE
71 -73 -79

31. How to solve: Is the integer z divisible by 6? (1) gcd(z -12) = 3 (2) gcd(z -15) = 15
61 -67
23 -29
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹
The PRODUCT of n consecutive integers is divisible by n!.

32. How to find the sum of consecutive integers:
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹
In an evenly spaced set - the average and the median are equal.
71 -73 -79
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.

33. ³v216 =
41 -43 -47
3·3n = 3^{n+1}
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6
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

34. Any integer with an EVEN number of total factors cannot be ______.
FACTOR
A PERFECT SQUARE
1. The smallest or largest element 2. The increment 3. The number of items in the set
Either a multiple of N or a non-multiple of N

35. Prime factors of _____ must come in pairs of three.
PERFECT CUBES
ODD
61 -67
A PERFECT SQUARE

36. v3˜
[(last - first) / increment] + 1
The average of an ODD number of consecutive integers will ALWAYS be an integer.
If 2 cannot be one of the primes in the sum - the sum must be even.
1.7

37. Let N be an integer. If you add a multiple of N to a non-multiple of N - the result is ________.
16
If 2 cannot be one of the primes in the sum - the sum must be even.
13
A non-multiple of N.

38. If estimating a root with a coefficient - _____ .
Prime
2.5
Put the coefficient under the radical to get a better approximation
25

39. Positive integers with only two factors must be ___.
Prime
2.5
53 -59
1.4

40. Prime Numbers:1x
11 -13 -17 -19
2 -3 -5 -7
The average of the set times the number of elements in the set
83 -89

41. v225=
15
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.
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.
23 -29

42. Positive integers with more than two factors are ____.
2 -3 -5 -7
1.4
Never prime
A PERFECT SQUARE

43. v169=
13
The average of an ODD number of consecutive integers will ALWAYS be an integer.
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹

44. If the problem states/assumes that a number is an integer - check to see if you can use _______.
The average of an EVEN number of consecutive integers will NEVER be an integer.
71 -73 -79
Prime factorization
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.

45. 3n + 3n + 3n = _____ = ______
1.4
NEVER CONTRADICT ONE ANOTHER
3·3n = 3^{n+1}
13

46. On data sufficiency - ALWAYS _______ algebraic expressions when you can. ESPECIALLY for divisibility.
FACTOR
Prime
1.4
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6

47. In an evenly spaced set - the sum of the terms is equal to ____.
53 -59
The average of the set times the number of elements in the set
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.
23 -29

48. 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?
Never prime
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6
3·3n = 3^{n+1}
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.

49. Prime Numbers:6x
The average of an ODD number of consecutive integers will ALWAYS be an integer.
The average of an EVEN number of consecutive integers will NEVER be an integer.
The sum of any two primes will be even - unless one of the two primes is 2.
61 -67

50. The SUM of n consecutive integers is divisible by n if ____ - but not if ______.
ODD
The average of an ODD number of consecutive integers will ALWAYS be an integer.
1.7
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.