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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. Any integer with an ODD number of total factors must be _______.
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.
A PERFECT SQUARE
16
13

2. If the problem states/assumes that a number is an integer - check to see if you can use _______.
Prime factorization
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.
14
Either a multiple of N or a non-multiple of N

3. On data sufficiency - ALWAYS _______ algebraic expressions when you can. ESPECIALLY for divisibility.
FACTOR
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.
Never prime
71 -73 -79

4. 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
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.
Either a multiple of N or a non-multiple of N
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹

5. v225=
14
83 -89
The average of the set times the number of elements in the set
15

6. Positive integers with only two factors must be ___.
The sum of any two primes will be even - unless one of the two primes is 2.
PERFECT CUBES
2 -3 -5 -7
Prime

7. Prime Numbers:8x
83 -89
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.

8. v256=
EVEN
1. The smallest or largest element 2. The increment 3. The number of items in the set
16
53 -59

9. 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.
EVEN
The average of an EVEN number of consecutive integers will NEVER be an integer.
31 -37

10. Prime Numbers:9x
53 -59
Prime factorization
97
The middle number

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

12. Prime Numbers:0x
A PERFECT SQUARE
2 -3 -5 -7
41 -43 -47
The middle number

13. In an evenly spaced set - the sum of the terms is equal to ____.
NEVER CONTRADICT ONE ANOTHER
1.7
The average of the set times the number of elements in the set
2 -3 -5 -7

14. The sum of any two primes will be ____ - unless ______.
14
The sum of any two primes will be even - unless one of the two primes is 2.
NEVER CONTRADICT ONE ANOTHER
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹

15. Prime Numbers:6x
A PERFECT SQUARE
15
61 -67
FACTOR

16. The prime factorization of __________ contains only EVEN powers of primes.
97
53 -59
A PERFECT SQUARE
1.7

17. The formula for finding the number of consecutive multiples in a set is _______.
The middle number
[(last - first) / increment] + 1
A non-multiple of N.
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6

18. N! is _____ of all integers from 1 to N.
A MULTIPLE
1. The smallest or largest element 2. The increment 3. The number of items in the set
71 -73 -79
83 -89

19. v169=
N is a divisor of x+y
3·3n = 3^{n+1}
13
14

20. 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
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
The middle number
11 -13 -17 -19
N is a divisor of x+y

21. Let N be an integer. If you add two non-multiples of N - the result could be _______.
Either a multiple of N or a non-multiple of N
Prime factorization
2.5
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹

22. v196=
97
FACTOR
14
The sum of any two primes will be even - unless one of the two primes is 2.

23. v625=
25
In an evenly spaced set - the average and the median are equal.
97
14

24. v3˜
1.7
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.
The PRODUCT of n consecutive integers is divisible by n!.
25

25. 3n + 3n + 3n = _____ = ______
In an evenly spaced set - the average and the median are equal.
3·3n = 3^{n+1}
Never prime
14

26. Prime Numbers:5x
53 -59
1.4
The same sign as the base
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.

27. ³v216 =
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.
[(last - first) / increment] + 1
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6
1.4

28. How to solve: Is the integer z divisible by 6? (1) gcd(z -12) = 3 (2) gcd(z -15) = 15
13
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.
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹

29. The PRODUCT of n consecutive integers is divisible by ____.
61 -67
If 2 cannot be one of the primes in the sum - the sum must be even.
The PRODUCT of n consecutive integers is divisible by n!.
Prime factorization

30. v2˜
61 -67
1.4
The middle number
FACTOR

31. The two statements in a data sufficiency problem will _______________.
53 -59
Never prime
NEVER CONTRADICT ONE ANOTHER
PERFECT CUBES

32. Prime factors of _____ must come in pairs of three.
If 2 cannot be one of the primes in the sum - the sum must be even.
[(last - first) / increment] + 1
PERFECT CUBES
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

33. 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.
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.
11 -13 -17 -19

34. When we take an EVEN ROOT - a radical sign means ________. This is _____ even exponents.
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
13
1.4
ONLY the nonnegative root of the numberUNLIKE

35. The SUM of n consecutive integers is divisible by n if ____ - but not if ______.
1.4
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.
23 -29
15

36. Prime Numbers:1x
A non-multiple of N.
11 -13 -17 -19
[(last - first) / increment] + 1
The same sign as the base

37. For ODD ROOTS - the root has ______.
The same sign as the base
A PERFECT SQUARE
83 -89
Put the coefficient under the radical to get a better approximation

38. How to find the sum of consecutive integers:
A PERFECT SQUARE
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.
71 -73 -79
1. The smallest or largest element 2. The increment 3. The number of items in the set

39. All perfect squares have a(n) _________ number of total factors.
41 -43 -47
ODD
31 -37
23 -29

40. If estimating a root with a coefficient - _____ .
14
97
FACTOR
Put the coefficient under the radical to get a better approximation

41. Prime Numbers:2x
23 -29
A MULTIPLE
3·3n = 3^{n+1}
The middle number

42. v5˜
The PRODUCT of n consecutive integers is divisible by n!.
2.5
ODD
83 -89

43. Prime Numbers:7x
ODD
71 -73 -79
53 -59
A PERFECT SQUARE

44. Any integer with an EVEN number of total factors cannot be ______.
If 2 cannot be one of the primes in the sum - the sum must be even.
83 -89
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹
A PERFECT SQUARE

45. The average of an ODD number of consecutive integers will ________ be an integer.
The average of an ODD number of consecutive integers will ALWAYS be an integer.
13
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.
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹

46. The average of an EVEN number of consecutive integers will ________ be an integer.
83 -89
In an evenly spaced set - the average and the median are equal.
11 -13 -17 -19
The average of an EVEN number of consecutive integers will NEVER be an integer.

47. Prime Numbers:4x
The average of the set times the number of elements in the set
2 -3 -5 -7
NEVER CONTRADICT ONE ANOTHER
41 -43 -47

48. In an evenly spaced set - the ____ and the ____ are equal.
ODD
Prime
Put the coefficient under the radical to get a better approximation
In an evenly spaced set - the average and the median are equal.

49. Let N be an integer. If you add a multiple of N to a non-multiple of N - the result is ________.
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.
The same sign as the base
A non-multiple of N.
1.7

50. Prime Numbers:3x
31 -37
N is a divisor of x+y
ONLY the nonnegative root of the numberUNLIKE
EVEN