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25 Mar 2013, 04:50
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Number Properties is a key topic on the GMAT and it's important to have a solid understanding of it. You'll encounter questions that test your ability to work with integers, multiples and factors, primes, odds and evens, and more. In this set of GMAT questions from GMAT Club Tests, you'll face 11 challenging problems that will put your number properties knowledge to the test. Good luck!
1. If x is an integer, what is the value of x?
(1) |23x| is a prime number
(2) \(2\sqrt{x^2}\) is a prime number.
2. If a positive integer n has exactly two positive factors what is the value of n?
(1) n/2 is one of the factors of n
(2) The lowest common multiple of n and n + 10 is an even number.
3. If 0 < x < y and x and y are consecutive perfect squares, what is the remainder when y is divided by x?
(1) Both x and y is have 3 positive factors.
(2) Both \(\sqrt{x}\) and \(\sqrt{y}\) are prime numbers
4. If each digit of the positive three-digit integer \(k\) is a positive multiple of 4, what is the value of \(k\)?
(1) The units digit of \(k\) is the least common multiple of the tens and hundreds digits of \(k\).
(2) \(k\) is NOT a multiple of 3.
5. If a, b, and c are integers and a < b < c, are a, b, and c consecutive integers?
(1) The median of {a!, b!, c!} is an odd number.
(2) c! is a prime number
6. Set S consists of more than two integers. Are all the numbers in set S negative?
(1) The product of any three integers in the list is negative
(2) The product of the smallest and largest integers in the list is a prime number.
7. Is x the square of an integer?
(1) When x is divided by 12 the remainder is 6
(2) When x is divided by 14 the remainder is 2
8. List A consist of 10 terms, each of which is a reciprocal of a prime number, is the median of the list less than 1/5?
(1) Reciprocal of the median is a prime number
(2) The product of any two terms of the list is a terminating decimal
9. If [x] denotes the greatest integer less than or equal to x for any number x, is [a] + [b] = 1 ?
(1) ab = 2
(2) 0 < a < b < 2
10. If N = 3^x*5^y, where x and y are positive integers, and N has 12 positive factors, what is the value of N?
(1) 9 is NOT a factor of N
(2) 125 is a factor of N
BONUS QUESTION:
11. If x and y are positive integers, is x a prime number?
(1) |x - 2| < 2 - y
(2) x + y - 3 = |1-y|
The solutions can be found below on this page.
1. If x is an integer, what is the value of x?
(1) |23x| is a prime number
(2) \(2\sqrt{x^2}\) is a prime number.
2. If a positive integer n has exactly two positive factors what is the value of n?
(1) n/2 is one of the factors of n
(2) The lowest common multiple of n and n + 10 is an even number.
3. If 0 < x < y and x and y are consecutive perfect squares, what is the remainder when y is divided by x?
(1) Both x and y is have 3 positive factors.
(2) Both \(\sqrt{x}\) and \(\sqrt{y}\) are prime numbers
4. If each digit of the positive three-digit integer \(k\) is a positive multiple of 4, what is the value of \(k\)?
(1) The units digit of \(k\) is the least common multiple of the tens and hundreds digits of \(k\).
(2) \(k\) is NOT a multiple of 3.
5. If a, b, and c are integers and a < b < c, are a, b, and c consecutive integers?
(1) The median of {a!, b!, c!} is an odd number.
(2) c! is a prime number
6. Set S consists of more than two integers. Are all the numbers in set S negative?
(1) The product of any three integers in the list is negative
(2) The product of the smallest and largest integers in the list is a prime number.
7. Is x the square of an integer?
(1) When x is divided by 12 the remainder is 6
(2) When x is divided by 14 the remainder is 2
8. List A consist of 10 terms, each of which is a reciprocal of a prime number, is the median of the list less than 1/5?
(1) Reciprocal of the median is a prime number
(2) The product of any two terms of the list is a terminating decimal
9. If [x] denotes the greatest integer less than or equal to x for any number x, is [a] + [b] = 1 ?
(1) ab = 2
(2) 0 < a < b < 2
10. If N = 3^x*5^y, where x and y are positive integers, and N has 12 positive factors, what is the value of N?
(1) 9 is NOT a factor of N
(2) 125 is a factor of N
BONUS QUESTION:
11. If x and y are positive integers, is x a prime number?
(1) |x - 2| < 2 - y
(2) x + y - 3 = |1-y|
The solutions can be found below on this page.
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29 Mar 2013, 05:34
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8. List A consist of 10 terms, each of which is a reciprocal of a prime number, is the median of the list less than 1/5?
(1) Reciprocal of the median is a prime number. If all the terms equal 1/2, then the median=1/2 and the answer is NO but if all the terms equal 1/7, then the median=1/7 and the answer is YES. Not sufficient.
(2) The product of any two terms of the list is a terminating decimal. This statement implies that the list must consists of 1/2 or/and 1/5. Thus the median could be 1/2, 1/5 or (1/5+1/2)/2=7/20. None of the possible values is less than 1/5. Sufficient.
Answer: B.
Theory:
Reduced fraction \(\frac{a}{b}\) (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only \(b\) (denominator) is of the form \(2^n5^m\), where \(m\) and \(n\) are non-negative integers. For example: \(\frac{7}{250}\) is a terminating decimal \(0.028\), as \(250\) (denominator) equals to \(2*5^3\). Fraction \(\frac{3}{30}\) is also a terminating decimal, as \(\frac{3}{30}=\frac{1}{10}\) and denominator \(10=2*5\).
Note that if denominator already has only 2-s and/or 5-s then it doesn't matter whether the fraction is reduced or not.
For example \(\frac{x}{2^n5^m}\), (where x, n and m are integers) will always be the terminating decimal.
We need reducing in case when we have the prime in denominator other then 2 or 5 to see whether it could be reduced. For example fraction \(\frac{6}{15}\) has 3 as prime in denominator and we need to know if it can be reduced.
Questions testing this concept:
https://gmatclub.com/forum/does-the-deci ... 89566.html
https://gmatclub.com/forum/any-decimal-t ... 01964.html
https://gmatclub.com/forum/if-a-b-c-d-an ... 25789.html
https://gmatclub.com/forum/700-question-94641.html
https://gmatclub.com/forum/is-r-s2-is-a- ... 91360.html
https://gmatclub.com/forum/pl-explain-89566.html
https://gmatclub.com/forum/which-of-the- ... 88937.html
(1) Reciprocal of the median is a prime number. If all the terms equal 1/2, then the median=1/2 and the answer is NO but if all the terms equal 1/7, then the median=1/7 and the answer is YES. Not sufficient.
(2) The product of any two terms of the list is a terminating decimal. This statement implies that the list must consists of 1/2 or/and 1/5. Thus the median could be 1/2, 1/5 or (1/5+1/2)/2=7/20. None of the possible values is less than 1/5. Sufficient.
Answer: B.
Theory:
Reduced fraction \(\frac{a}{b}\) (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only \(b\) (denominator) is of the form \(2^n5^m\), where \(m\) and \(n\) are non-negative integers. For example: \(\frac{7}{250}\) is a terminating decimal \(0.028\), as \(250\) (denominator) equals to \(2*5^3\). Fraction \(\frac{3}{30}\) is also a terminating decimal, as \(\frac{3}{30}=\frac{1}{10}\) and denominator \(10=2*5\).
Note that if denominator already has only 2-s and/or 5-s then it doesn't matter whether the fraction is reduced or not.
For example \(\frac{x}{2^n5^m}\), (where x, n and m are integers) will always be the terminating decimal.
We need reducing in case when we have the prime in denominator other then 2 or 5 to see whether it could be reduced. For example fraction \(\frac{6}{15}\) has 3 as prime in denominator and we need to know if it can be reduced.
Questions testing this concept:
https://gmatclub.com/forum/does-the-deci ... 89566.html
https://gmatclub.com/forum/any-decimal-t ... 01964.html
https://gmatclub.com/forum/if-a-b-c-d-an ... 25789.html
https://gmatclub.com/forum/700-question-94641.html
https://gmatclub.com/forum/is-r-s2-is-a- ... 91360.html
https://gmatclub.com/forum/pl-explain-89566.html
https://gmatclub.com/forum/which-of-the- ... 88937.html
29 Mar 2013, 06:11
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Bookmarks
11. If x and y are positive integers, is x a prime number?
(1) |x - 2| < 2 - y . The left hand side of the inequality is an absolute value, so the least value of LHS is zero, thus 0 < 2 - y, thus y < 2 (if y is more than or equal to 2, then \(2-y\leq{0}\) and it cannot be greater than |x - 2|). Next, since given that y is a positive integer, then y=1.
So, we have that: \(|x - 2| < 1\), which implies that \(-1 < x-2 < 1\), or \(1 < x < 3\), thus \(x=2=prime\). Sufficient.
(2) x + y - 3 = |1-y|. Since y is a positive integer, then \(1-y\leq{0}\), thus \(|1-y|=-(1-y)\). So, we have that \(x + y - 3 = -(1-y)\), which gives \(x=2=prime\). Sufficient.
Answer: D.
(1) |x - 2| < 2 - y . The left hand side of the inequality is an absolute value, so the least value of LHS is zero, thus 0 < 2 - y, thus y < 2 (if y is more than or equal to 2, then \(2-y\leq{0}\) and it cannot be greater than |x - 2|). Next, since given that y is a positive integer, then y=1.
So, we have that: \(|x - 2| < 1\), which implies that \(-1 < x-2 < 1\), or \(1 < x < 3\), thus \(x=2=prime\). Sufficient.
(2) x + y - 3 = |1-y|. Since y is a positive integer, then \(1-y\leq{0}\), thus \(|1-y|=-(1-y)\). So, we have that \(x + y - 3 = -(1-y)\), which gives \(x=2=prime\). Sufficient.
Answer: D.
