Number Properties

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ela07mjt: Junior | Next Rank: 30 Posts; Posts: 20; Joined: Wed Oct 31, 2012 3:53 am

Number Properties

by ela07mjt » Tue Feb 19, 2013 8:38 am

1. Is the positive integer j divisible by a greater number of different prime numbers than the positive integer k ?
(1) j is divisible by 30.
(2) k = 1,000

2. If m is a positive odd integer between 2 and 30, then m is divisible by how many different positive prime numbers?
(1) m is not divisible by 3.
(2) m is not divisible by 5.

3. If k is an integer greater than 1, is k equal to 2r for some positive integer r ?
(1) k is divisible by 26.
(2) k is not divisible by any odd integer greater than 1.

4. If n is a positive integer and r is the remainder when n2 - 1 is divided by 8, what is the value of r ?
(1) n is odd.
(2) n is not divisible by 8.

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Source: Beat The GMAT — Data Sufficiency | Tue Feb 19, 2013 8:38 am

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pm; Location: Milpitas, CA; Thanked: 1854 times; Followed by:523 members; GMAT Score:770

by Anurag@Gurome » Tue Feb 19, 2013 8:56 am

ela07mjt wrote:Is the positive integer j divisible by a greater number of different prime numbers than the positive integer k ?

(1) j is divisible by 30.
(2) k = 1,000

Statement 1: We don't know anything about k ---> Not sufficient

Statement 2: We don't know anything about j ---> Not sufficient

1 & 2 Together: As 30 = 2*3*5, j is divisible by at least 3 different prime numbers. Whereas k = 1000 = (2*5)^3 is divisible by only two different prime numbers. Hence, we can definitely answer the originally question in YES.

Sufficient
[spoiler]
The correct answer is C.[/spoiler]

Last edited by Anurag@Gurome on Tue Feb 19, 2013 8:59 am, edited 1 time in total.

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 am; Location: Montreal; Thanked: 1090 times; Followed by:355 members; GMAT Score:780

by Ian Stewart » Tue Feb 19, 2013 8:57 am

ela07mjt wrote:1. Is the positive integer j divisible by a greater number of different prime numbers than the positive integer k ?
(1) j is divisible by 30.
(2) k = 1,000

We need information about both j and k, so neither statement is sufficient alone. From Statement 2, we know k = (2^3)(5^3), so k is divisible by exactly two distinct primes, while from Statement 1 we know j is divisible by 2*3*5, so is divisible by at least 3 distinct primes. So the answer is C.

ela07mjt wrote: 2. If m is a positive odd integer between 2 and 30, then m is divisible by how many different positive prime numbers?
(1) m is not divisible by 3.
(2) m is not divisible by 5.

m is odd. If m is not divisible by 3, and m has two distinct prime divisors, the smallest possible value of m is 5*7 = 35 (the product of the two smallest odd primes other than 3). But that's too large, since m < 30. So using Statement 1, m can only have one prime divisor, and Statement 1 is sufficient. Using Statement 2, m could have only one prime divisor, but m could also be equal to 3*7 = 21, and have two prime divisors. So the answer is A.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 am; Location: Montreal; Thanked: 1090 times; Followed by:355 members; GMAT Score:780

by Ian Stewart » Tue Feb 19, 2013 9:07 am

ela07mjt wrote:
3. If k is an integer greater than 1, is k equal to 2r for some positive integer r ?
(1) k is divisible by 26.
(2) k is not divisible by any odd integer greater than 1.

If k = 2r, then k is even, so the question is just asking if k is an even number. If k is divisible by 26, then k is certainly divisible by 2, so k is even, and Statement 1 is sufficient. If k > 1, and k has no odd divisors besides 1, then the only prime divisor of k must be 2. So using Statement 2, we know k = 2^n (k is some power of 2), and k is even. So the answer is D.

The question is badly constructed, however, since it's impossible for both statements to be true. I wonder if Statement 1 is supposed to read "k is divisible by 2^6 ?"

ela07mjt wrote:
4. If n is a positive integer and r is the remainder when n2 - 1 is divided by 8, what is the value of r ?
(1) n is odd.
(2) n is not divisible by 8.

Statement 2 is not sufficient, as you can see by plugging in, say, n=4 and n=1. Statement 1 is sufficient. You can see that in a few ways; one is to just plug in the four odd values 1, 3, 5 and 7 (the four different odd remainders you can have when you divide by 8), and you will see that n^2 - 1 is always divisible by 8. Or you can notice that n^2 - 1 = (n+1)(n-1). If n is odd, then n-1 and n+1 are consecutive even numbers, and if you take any two consecutive even numbers, one of them will always be a multiple of 4. So one of n-1 or n+1 is divisible by 4, and the other is divisible by 2, so their product n^2 - 1 must be divisible by 8.

The longest approach is to proceed algebraically, but that will also work. If n is odd, n = 2k+1 for some integer k. So

n^2 - 1 = (2k + 1)^2 - 1
= 4k^2 + 4k + 1 - 1
= 4(k^2 + k)
= 4(k)(k+1)

Notice that one of k or k+1 must be even, since k and k+1 are consecutive integers, so 4(k)(k+1) must be divisible by 8. So using Statement 1, the remainder is 0 when n^2 - 1 is divided by 8.

Last edited by Ian Stewart on Tue Feb 19, 2013 9:10 am, edited 1 time in total.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by Anurag@Gurome » Tue Feb 19, 2013 9:09 am

ela07mjt wrote:3. If k is an integer greater than 1, is k equal to 2r for some positive integer r ?
(1) k is divisible by 26.
(2) k is not divisible by any odd integer greater than 1.

The question is simply asking whether k is even or not.

Statement 1: k is divisible by 26, hence k is even ---> Sufficient

Statement 2: As k is an integer greater than 1 but not divisible by any odd integer greater than 1, k must be an even integer ---> Sufficient

The correct answer is D.

Note : In proper GMAT DS questions the statements do not contradict each other. However, here from 1st statement k is clearly divisible by 13 but from 2nd it is not. This is not a proper DS question.