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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: 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

2. 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.
PERFECT CUBES
Never prime
A MULTIPLE

3. In an evenly spaced set - the average can be found by finding ________.
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.
2.5
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.
The middle number

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

5. Prime Numbers:6x
The sum of any two primes will be even - unless one of the two primes is 2.
61 -67
ODD
23 -29

6. Prime Numbers:7x
A MULTIPLE
A non-multiple of N.
53 -59
71 -73 -79

7. Prime Numbers:5x
53 -59
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.
NEVER CONTRADICT ONE ANOTHER
1. The smallest or largest element 2. The increment 3. The number of items in the set

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

9. Prime Numbers:8x
83 -89
23 -29
PERFECT CUBES
25

10. v3˜
1.7
The middle number
Prime
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.

11. 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?
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.
A MULTIPLE
In an evenly spaced set - the average and the median are equal.

12. The prime factorization of __________ contains only EVEN powers of primes.
A PERFECT SQUARE
97
NEVER CONTRADICT ONE ANOTHER
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6

13. In an evenly spaced set - the sum of the terms is equal to ____.
The average of the set times the number of elements in the set
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.
3·3n = 3^{n+1}
13

14. If estimating a root with a coefficient - _____ .
The PRODUCT of n consecutive integers is divisible by n!.
Put the coefficient under the radical to get a better approximation
53 -59
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6

15. Prime Numbers:3x
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.
31 -37
Prime
PERFECT CUBES

16. Prime Numbers:0x
2 -3 -5 -7
N is a divisor of x+y
The sum of any two primes will be even - unless one of the two primes is 2.
The average of an ODD number of consecutive integers will ALWAYS be an integer.

17. v625=
ONLY the nonnegative root of the numberUNLIKE
Prime factorization
25
If 2 cannot be one of the primes in the sum - the sum must be even.

18. In an evenly spaced set - the ____ and the ____ are equal.
41 -43 -47
3·3n = 3^{n+1}
In an evenly spaced set - the average and the median 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.

19. Positive integers with only two factors must be ___.
NEVER CONTRADICT ONE ANOTHER
Prime
A MULTIPLE
EVEN

20. Let N be an integer. If you add a multiple of N to a non-multiple of N - the result is ________.
16
A non-multiple of N.
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.

21. The two statements in a data sufficiency problem will _______________.
1.7
31 -37
The sum of any two primes will be even - unless one of the two primes is 2.
NEVER CONTRADICT ONE ANOTHER

22. The PRODUCT of n consecutive integers is divisible by ____.
EVEN
The PRODUCT of n consecutive integers is divisible by n!.
1.7
2 -3 -5 -7

23. The sum of any two primes will be ____ - unless ______.
16
A PERFECT SQUARE
If 2 cannot be one of the primes in the sum - the sum must be even.
The sum of any two primes will be even - unless one of the two primes is 2.

24. Prime Numbers:9x
97
[(last - first) / increment] + 1
The SUM of n consecutive integers is divisible by n if n is odd - but not if n is even.
Break the number into prime powers: 216 = 2 2 2 3 3 * 3 = 2³ · 3³ = 6³ - so ³v216 = ³v6³ = 6

25. N! is _____ of all integers from 1 to N.
The average of an EVEN number of consecutive integers will NEVER be an integer.
ODD
A PERFECT SQUARE
A MULTIPLE

26. The average of an ODD number of consecutive integers will ________ be an integer.
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 average of an ODD number of consecutive integers will ALWAYS be an integer.
53 -59
The PRODUCT of n consecutive integers is divisible by n!.

27. The formula for finding the number of consecutive multiples in a set is _______.
[(last - first) / increment] + 1
61 -67
14
PERFECT CUBES

28. In an evenly spaced set - the mean and median are equal to the _____ of _________.
Prime
31 -37
In an evenly spaced set - the mean and median are equal to the average of the first and the last number.
In an evenly spaced set - the average and the median are equal.

29. 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
Never prime
25
83 -89
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

30. For ODD ROOTS - the root has ______.
31 -37
Put the coefficient under the radical to get a better approximation
11 -13 -17 -19
The same sign as the base

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

32. If N is a divisor of x and y - then _______.
53 -59
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
[(last - first) / increment] + 1

33. v5˜
ODD
53 -59
2.5
If 2 cannot be one of the primes in the sum - the sum must be even.

34. v225=
15
A non-multiple of N.
A PERFECT SQUARE
A PERFECT SQUARE

35. v2˜
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.
1.4
NEVER CONTRADICT ONE ANOTHER

36. When we take an EVEN ROOT - a radical sign means ________. This is _____ even exponents.
11 -13 -17 -19
A MULTIPLE
The PRODUCT of n consecutive integers is divisible by n!.
ONLY the nonnegative root of the numberUNLIKE

37. On data sufficiency - ALWAYS _______ algebraic expressions when you can. ESPECIALLY for divisibility.
FACTOR
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹
PERFECT CUBES
1. The smallest or largest element 2. The increment 3. The number of items in the set

38. 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
23 -29
N is a divisor of x+y
15
If gcd(k1 -n) ? 1 or gcd(k2 -n) ? 1 - this proves insufficiency.

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

40. How to solve: Is the integer z divisible by 6? (1) gcd(z -12) = 3 (2) gcd(z -15) = 15
2 -3 -5 -7
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹
53 -59
41 -43 -47

41. v169=
The sum of any two primes will be even - unless one of the two primes is 2.
13
The average of an EVEN number of consecutive integers will NEVER be an integer.
3·3n = 3^{n+1}

42. Prime Numbers:4x
The PRODUCT of n consecutive integers is divisible by n!.
41 -43 -47
61 -67
14

43. How to find the sum of consecutive integers:
14
23 -29
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 average of an EVEN number of consecutive integers will NEVER be an integer.

44. v196=
The sum of any two primes will be even - unless one of the two primes is 2.
61 -67
14
16

45. If 2 cannot be one of the primes in the sum - the sum must be _____.
53 -59
83 -89
Set up prime columns. -- z 6 12 15 2 --2¹ 2² 3 --3¹ 3¹ 3¹ 5 ---------5¹
If 2 cannot be one of the primes in the sum - the sum must be even.

46. Prime Numbers:2x
11 -13 -17 -19
The average of an EVEN number of consecutive integers will NEVER be an integer.
The average of an ODD number of consecutive integers will ALWAYS be an integer.
23 -29

47. v256=
If 2 cannot be one of the primes in the sum - the sum must be even.
N is a divisor of x+y
A PERFECT SQUARE
16

48. Positive integers with more than two factors are ____.
If 2 cannot be one of the primes in the sum - the sum must be even.
Never prime
The same sign as the base
Either a multiple of N or a non-multiple of N

49. 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.
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
11 -13 -17 -19

50. All perfect squares have a(n) _________ number of total factors.
FACTOR
A PERFECT SQUARE
ODD
The same sign as the base