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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. Prime factors of _____ must come in pairs of three.
A PERFECT SQUARE
PERFECT CUBES
Prime factorization
2 -3 -5 -7

2. Prime Numbers:3x
41 -43 -47
25
31 -37
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.

3. If estimating a root with a coefficient - _____ .
Put the coefficient under the radical to get a better approximation
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.
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.
53 -59

4. N! is _____ of all integers from 1 to N.
1.7
A MULTIPLE
The middle number
The average of an EVEN number of consecutive integers will NEVER be an integer.

5. 3n + 3n + 3n = _____ = ______
ODD
Either a multiple of N or a non-multiple of N
33n = 3^{n+1}
13

6. 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?
14
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.
1.7
1. The smallest or largest element 2. The increment 3. The number of items in the set

7. The formula for finding the number of consecutive multiples in a set is _______.
71 -73 -79
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.
11 -13 -17 -19
[(last - first) / increment] + 1

8. How to find the sum of consecutive integers:
EVEN
N is a divisor of x+y
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2 3 = 6 - so v216 = v6 = 6
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.

9. v256=
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.
16
15
41 -43 -47

10. v625=
25
The middle number
1.7
Set up prime columns. -- z 6 12 15 2 --2 2 3 --3 3 3 5 ---------5

11. v225=
The average of an ODD number of consecutive integers will ALWAYS be an integer.
15
2 -3 -5 -7
FACTOR

12. Prime Numbers:7x
2.5
Either a multiple of N or a non-multiple of N
2 -3 -5 -7
71 -73 -79

13. Any integer with an ODD number of total factors must be _______.
A PERFECT SQUARE
The average of an EVEN number of consecutive integers will NEVER be an integer.
In an evenly spaced set - the average and the median are equal.
Set up prime columns. -- z 6 12 15 2 --2 2 3 --3 3 3 5 ---------5

14. If 2 cannot be one of the primes in the sum - the sum must be _____.
The same sign as the base
ONLY the nonnegative root of the numberUNLIKE
If 2 cannot be one of the primes in the sum - the sum must be even.
41 -43 -47

15. In an evenly spaced set - the ____ and the ____ are equal.
83 -89
The middle number
The sum of any two primes will be even - unless one of the two primes is 2.
In an evenly spaced set - the average and the median are equal.

16. Positive integers with only two factors must be ___.
41 -43 -47
71 -73 -79
Prime
83 -89

17. 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
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.
[(last - first) / increment] + 1
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.

18. The average of an ODD number of consecutive integers will ________ be an integer.
Set up prime columns. -- z 6 12 15 2 --2 2 3 --3 3 3 5 ---------5
The average of an ODD number of consecutive integers will ALWAYS be an integer.
1. The smallest or largest element 2. The increment 3. The number of items in the set
11 -13 -17 -19

19. Prime Numbers:6x
61 -67
[(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
The same sign as the base

20. Prime Numbers:4x
The middle number
If 2 cannot be one of the primes in the sum - the sum must be even.
31 -37
41 -43 -47

21. v5
2.5
61 -67
The sum of any two primes will be even - unless one of the two primes is 2.
1.4

22. 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
2 -3 -5 -7
The average of an ODD number of consecutive integers will ALWAYS be an integer.
Either a multiple of N or a non-multiple of N

23. Let N be an integer. If you add two non-multiples of N - the result could be _______.
The average of the set times the number of elements in the set
Either a multiple of N or a non-multiple of N
The middle number
Never prime

24. If the problem states/assumes that a number is an integer - check to see if you can use _______.
31 -37
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.
83 -89
Prime factorization

25. For ODD ROOTS - the root has ______.
1.7
PERFECT CUBES
83 -89
The same sign as the base

26. All perfect squares have a(n) _________ number of total factors.
The middle number
2 -3 -5 -7
ODD
33n = 3^{n+1}

27. In an evenly spaced set - the sum of the terms is equal to ____.
Put the coefficient under the radical to get a better approximation
The average of the set times the number of elements in the set
97
A non-multiple of N.

28. v169=
ONLY the nonnegative root of the numberUNLIKE
Set up prime columns. -- z 6 12 15 2 --2 2 3 --3 3 3 5 ---------5
13
The average of an EVEN number of consecutive integers will NEVER be an integer.

29. Any integer with an EVEN number of total factors cannot be ______.
A PERFECT SQUARE
If 2 cannot be one of the primes in the sum - the sum must be even.
11 -13 -17 -19
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.

30. If N is a divisor of x and y - then _______.
N is a divisor of x+y
41 -43 -47
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.
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.

31. v196=
41 -43 -47
14
11 -13 -17 -19
Put the coefficient under the radical to get a better approximation

32. When we take an EVEN ROOT - a radical sign means ________. This is _____ even exponents.
A MULTIPLE
ONLY the nonnegative root of the numberUNLIKE
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.
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. Let N be an integer. If you add a multiple of N to a non-multiple of N - the result is ________.
A non-multiple of N.
In an evenly spaced set - the average and the median are equal.
23 -29
PERFECT CUBES

34. v3
15
PERFECT CUBES
11 -13 -17 -19
1.7

35. Prime Numbers:9x
The same sign as the base
Either a multiple of N or a non-multiple of N
97
If 2 cannot be one of the primes in the sum - the sum must be even.

36. 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.
Set up prime columns. -- z 6 12 15 2 --2 2 3 --3 3 3 5 ---------5
31 -37
EVEN

37. In an evenly spaced set - the average can be found by finding ________.
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.
The middle number
33n = 3^{n+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

38. Prime Numbers:2x
25
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.
13
23 -29

39. 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 --2 2 3 --3 3 3 5 ---------5
15
ODD
A PERFECT SQUARE

40. On data sufficiency - ALWAYS _______ algebraic expressions when you can. ESPECIALLY for divisibility.
A non-multiple of N.
53 -59
83 -89
FACTOR

41. In an evenly spaced set - the mean and median are equal to the _____ of _________.
33n = 3^{n+1}
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.
The PRODUCT of n consecutive integers is divisible by n!.
The average of the set times the number of elements in the set

42. The two statements in a data sufficiency problem will _______________.
NEVER CONTRADICT ONE ANOTHER
The middle number
FACTOR
16

43. The prime factorization of __________ contains only EVEN powers of primes.
14
N is a divisor of x+y
1.7
A PERFECT SQUARE

44. The average of an EVEN number of consecutive integers will ________ be an integer.
The average of an ODD number of consecutive integers will ALWAYS be an integer.
The middle number
The average of an EVEN number of consecutive integers will NEVER be an integer.
83 -89

45. Prime Numbers:1x
Prime
11 -13 -17 -19
A PERFECT SQUARE
A MULTIPLE

46. 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.
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.
The sum of any two primes will be even - unless one of the two primes is 2.

47. The prime factorization of a perfect square contains only ______ powers of primes.
Prime
EVEN
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.
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

48. Positive integers with more than two factors are ____.
Never prime
Prime
83 -89
71 -73 -79

49. v2
1.4
15
A PERFECT SQUARE
PERFECT CUBES

50. The PRODUCT of n consecutive integers is divisible by ____.
14
Set up prime columns. -- z 6 12 15 2 --2 2 3 --3 3 3 5 ---------5
Prime
The PRODUCT of n consecutive integers is divisible by n!.

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