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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.
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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. The average of an ODD number of consecutive integers will ________ be an integer.
EVEN
The average of an ODD number of consecutive integers will ALWAYS be an integer.
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6
Either a multiple of N or a non-multiple of N

2. Prime Numbers:8x
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.
97
ONLY the nonnegative root of the numberUNLIKE
83 -89

3. The prime factorization of __________ contains only EVEN powers of primes.
71 -73 -79
In an evenly spaced set - the average and the median are equal.
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6
A PERFECT SQUARE

4. Prime Numbers:0x
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!.
2 -3 -5 -7
A PERFECT SQUARE

5. Prime Numbers:2x
23 -29
2.5
Prime factorization
The middle number

6. Positive integers with only two factors must be ___.
A PERFECT SQUARE
16
Prime
1.4

7. How to find the sum of consecutive integers:
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.
2 -3 -5 -7
71 -73 -79
14

8. 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
1.7
Never prime
41 -43 -47
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.

9. ³v216 =
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6
71 -73 -79
25
The average of an EVEN number of consecutive integers will NEVER be an integer.

10. The formula for finding the number of consecutive multiples in a set is _______.
[(last - first) / increment] + 1
A non-multiple of N.
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

11. Prime Numbers:1x
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!.
The sum of any two primes will be even - unless one of the two primes is 2.
11 -13 -17 -19

12. The PRODUCT of n consecutive integers is divisible by ____.
N is a divisor of x+y
The PRODUCT of n consecutive integers is divisible by n!.
2 -3 -5 -7
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.

13. Let N be an integer. If you add a multiple of N to a non-multiple of N - the result is ________.
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6
If 2 cannot be one of the primes in the sum - the sum must be even.
A non-multiple of N.
In an evenly spaced set - the average and the median are equal.

14. The SUM of n consecutive integers is divisible by n if ____ - but not if ______.
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.
The average of an ODD number of consecutive integers will ALWAYS be an integer.
11 -13 -17 -19
1.7

15. In an evenly spaced set - the average can be found by finding ________.
Never prime
The average of the set times the number of elements in the set
The middle number
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.

16. In an evenly spaced set - the ____ and the ____ are equal.
In an evenly spaced set - the average and the median are equal.
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6
1.7
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.

17. Any integer with an ODD number of total factors must be _______.
A PERFECT SQUARE
Never prime
NEVER CONTRADICT ONE ANOTHER
14

18. Any integer with an EVEN number of total factors cannot be ______.
Prime
NEVER CONTRADICT ONE ANOTHER
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
A PERFECT SQUARE

19. If estimating a root with a coefficient - _____ .
53 -59
Put the coefficient under the radical to get a better approximation
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

20. Prime Numbers:5x
2.5
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.
The average of an EVEN number of consecutive integers will NEVER be an integer.
53 -59

21. Prime factors of _____ must come in pairs of three.
PERFECT CUBES
ONLY the nonnegative root of the numberUNLIKE
31 -37
1.7

22. 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
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.
2.5
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. Prime Numbers:9x
97
A MULTIPLE
ONLY the nonnegative root of the numberUNLIKE
61 -67

24. v196=
1.4
The middle number
Put the coefficient under the radical to get a better approximation
14

25. If the problem states/assumes that a number is an integer - check to see if you can use _______.
71 -73 -79
Prime factorization
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.
[(last - first) / increment] + 1

26. Positive integers with more than two factors are ____.
3·3n = 3^{n+1}
Never prime
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.
31 -37

27. 3n + 3n + 3n = _____ = ______
3·3n = 3^{n+1}
A non-multiple of N.
41 -43 -47
61 -67

28. The two statements in a data sufficiency problem will _______________.
NEVER CONTRADICT ONE ANOTHER
In an evenly spaced set - the average and the median are equal.
71 -73 -79
Prime

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

30. If 2 cannot be one of the primes in the sum - the sum must be _____.
ONLY the nonnegative root of the numberUNLIKE
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.
1. The smallest or largest element 2. The increment 3. The number of items in the set
If 2 cannot be one of the primes in the sum - the sum must be even.

31. Prime Numbers:3x
16
Either a multiple of N or a non-multiple of N
31 -37
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6

32. Prime Numbers:7x
2 -3 -5 -7
PERFECT CUBES
71 -73 -79
FACTOR

33. In an evenly spaced set - the mean and median are equal to the _____ of _________.
16
The average of an ODD number of consecutive integers will ALWAYS be an integer.
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.
ODD

34. The prime factorization of a perfect square contains only ______ powers of primes.
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
EVEN
53 -59
If 2 cannot be one of the primes in the sum - the sum must be even.

35. Prime Numbers:4x
PERFECT CUBES
A MULTIPLE
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.
41 -43 -47

36. On data sufficiency - ALWAYS _______ algebraic expressions when you can. ESPECIALLY for divisibility.
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.
FACTOR
15
A MULTIPLE

37. The average of an EVEN number of consecutive integers will ________ be an integer.
In an evenly spaced set - the average and the median are equal.
41 -43 -47
The average of an EVEN number of consecutive integers will NEVER be an integer.
ONLY the nonnegative root of the numberUNLIKE

38. 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
Prime factorization
A PERFECT SQUARE
FACTOR
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

39. Let N be an integer. If you add two non-multiples of N - the result could be _______.
Prime factorization
EVEN
FACTOR
Either a multiple of N or a non-multiple of N

40. All perfect squares have a(n) _________ number of total factors.
1. The smallest or largest element 2. The increment 3. The number of items in the set
The middle number
ODD
2 -3 -5 -7

41. v625=
N is a divisor of x+y
[(last - first) / increment] + 1
25
Put the coefficient under the radical to get a better approximation

42. v3˜
14
1.7
Put the coefficient under the radical to get a better approximation
NEVER CONTRADICT ONE ANOTHER

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

44. The sum of any two primes will be ____ - unless ______.
The sum of any two primes will be even - unless one of the two primes is 2.
EVEN
71 -73 -79
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.

45. v169=
The average of the set times the number of elements in the set
A non-multiple of N.
If 2 cannot be one of the primes in the sum - the sum must be even.
13

46. N! is _____ of all integers from 1 to N.
A PERFECT SQUARE
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6
A MULTIPLE
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.

47. If N is a divisor of x and y - then _______.
N is a divisor of x+y
[(last - first) / increment] + 1
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
NEVER CONTRADICT ONE ANOTHER

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?
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.
PERFECT CUBES
41 -43 -47
The sum of any two primes will be even - unless one of the two primes is 2.

49. v5˜
2.5
A MULTIPLE
FACTOR
61 -67

50. When we take an EVEN ROOT - a radical sign means ________. This is _____ even exponents.
A MULTIPLE
ONLY the nonnegative root of the numberUNLIKE
23 -29
The sum of any two primes will be even - unless one of the two primes is 2.

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GMAT Number Properties

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