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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. 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
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.
N is a divisor of x+y
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
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹

2. All perfect squares have a(n) _________ number of total factors.
ODD
25
A MULTIPLE
3·3n = 3^{n+1}

3. v5˜
A PERFECT SQUARE
2.5
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

4. Prime factors of _____ must come in pairs of three.
16
PERFECT CUBES
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

5. 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
1.4
11 -13 -17 -19
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹

6. Prime Numbers:5x
NEVER CONTRADICT ONE ANOTHER
FACTOR
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.
53 -59

7. For ODD ROOTS - the root has ______.
The same sign as the base
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.
Prime

8. Any integer with an ODD number of total factors must be _______.
A PERFECT SQUARE
The average of an ODD number of consecutive integers will ALWAYS be an integer.
FACTOR
25

9. In an evenly spaced set - the average can be found by finding ________.
The average of an EVEN number of consecutive integers will NEVER be an integer.
The middle number
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.
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.

10. ³v216 =
Either a multiple of N or 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.
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6
A PERFECT SQUARE

11. v3˜
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.
Never prime
1.7
The same sign as the base

12. 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.
2 -3 -5 -7
61 -67
13

13. Prime Numbers:7x
1.4
71 -73 -79
ODD
23 -29

14. The prime factorization of __________ contains only EVEN powers of primes.
A PERFECT SQUARE
13
FACTOR
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.

15. Prime Numbers:2x
83 -89
14
23 -29
53 -59

16. 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 middle number
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

17. 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

18. The prime factorization of a perfect square contains only ______ powers of primes.
Prime
FACTOR
EVEN
PERFECT CUBES

19. 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.
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.
31 -37
3·3n = 3^{n+1}

20. In an evenly spaced set - the ____ and the ____ are equal.
25
In an evenly spaced set - the average and the median are equal.
53 -59
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.

21. v625=
EVEN
25
31 -37
61 -67

22. v196=
14
Prime factorization
N is a divisor of x+y
A PERFECT SQUARE

23. How to find the sum of consecutive integers:
83 -89
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 middle number
1.4

24. 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.
ONLY the nonnegative root of the numberUNLIKE
2 -3 -5 -7
1.4

25. Prime Numbers:6x
61 -67
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 mean and median are equal to the average of the first and the last number.
The average of an EVEN number of consecutive integers will NEVER be an integer.

26. v169=
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹
14
13
NEVER CONTRADICT ONE ANOTHER

27. If N is a divisor of x and y - then _______.
A PERFECT SQUARE
2 -3 -5 -7
11 -13 -17 -19
N is a divisor of x+y

28. Any integer with an EVEN number of total factors cannot be ______.
83 -89
A PERFECT SQUARE
11 -13 -17 -19
14

29. Prime Numbers:9x
97
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
13

30. On data sufficiency - ALWAYS _______ algebraic expressions when you can. ESPECIALLY for divisibility.
FACTOR
1. The smallest or largest element 2. The increment 3. The number of items in the set
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.
31 -37

31. If the problem states/assumes that a number is an integer - check to see if you can use _______.
Either a multiple of N or a non-multiple of N
13
Prime factorization
11 -13 -17 -19

32. In an evenly spaced set - the sum of the terms is equal to ____.
NEVER CONTRADICT ONE ANOTHER
71 -73 -79
The average of the set times the number of elements in the set
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6

33. If 2 cannot be one of the primes in the sum - the sum must be _____.
A non-multiple of N.
If 2 cannot be one of the primes in the sum - the sum must be even.
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.

34. 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
3·3n = 3^{n+1}
1.7
31 -37

35. Prime Numbers:8x
13
A PERFECT SQUARE
The sum of any two primes will be even - unless one of the two primes is 2.
83 -89

36. All evenly spaced sets are fully defined if:1. _____ 2. _____ 3. _____ are known.
In an evenly spaced set - the average and the median are equal.
1. The smallest or largest element 2. The increment 3. The number of items in the set
A PERFECT SQUARE
97

37. The two statements in a data sufficiency problem will _______________.
N is a divisor of x+y
Put the coefficient under the radical to get a better approximation
In an evenly spaced set - the average and the median are equal.
NEVER CONTRADICT ONE ANOTHER

38. 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.
Never prime
A PERFECT SQUARE
14

39. If estimating a root with a coefficient - _____ .
FACTOR
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6
Put the coefficient under the radical to get a better approximation

40. N! is _____ of all integers from 1 to N.
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.
FACTOR
A MULTIPLE

41. Prime Numbers:1x
[(last - first) / increment] + 1
The PRODUCT of n consecutive integers is divisible by n!.
53 -59
11 -13 -17 -19

42. Positive integers with only two factors must be ___.
1.4
FACTOR
The sum of any two primes will be even - unless one of the two primes is 2.
Prime

43. 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
If 2 cannot be one of the primes in the sum - the sum must be even.
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.
61 -67
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.

44. When we take an EVEN ROOT - a radical sign means ________. This is _____ even exponents.
ONLY the nonnegative root of the numberUNLIKE
EVEN
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹
Never prime

45. v256=
23 -29
16
FACTOR
The average of an EVEN number of consecutive integers will NEVER be an integer.

46. 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.
The same sign as the base
The middle number
3·3n = 3^{n+1}

47. v225=
EVEN
The average of an EVEN number of consecutive integers will NEVER be an integer.
15
The sum of any two primes will be even - unless one of the two primes is 2.

48. Positive integers with more than two factors are ____.
Never prime
The average of the set times the number of elements in the set
In an evenly spaced set - the average and the median are equal.
3·3n = 3^{n+1}

49. The PRODUCT of n consecutive integers is divisible by ____.
The PRODUCT of n consecutive integers is divisible by n!.
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
1.4

50. The formula for finding the number of consecutive multiples in a set is _______.
97
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.
[(last - first) / increment] + 1
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.