Prime number

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vinay1983: Legendary Member; Posts: 643; Joined: Wed Aug 14, 2013 4:27 amFollowed by: 7 members; Thanked: 48 times

by vinay1983 » Sun Sep 29, 2013 11:33 pm

If P is a prime number, then which of the following COULD be the sum of the factors of P?

1. 2P-1 2. P+1 3. P^2-1

A. 2 only
B. 2 and 3 only
C. 1 and 3 only
D. 1 only
E. 1,2 and 3

You can, for example never foretell what any one man will do, but you can say with precision what an average number will be up to!

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rakeshd347: Master | Next Rank: 500 Posts; Posts: 391; Joined: Sat Mar 02, 2013 5:13 amFollowed by: 4 members; Thanked: 50 times

by rakeshd347 » Sun Sep 29, 2013 11:42 pm

vinay1983 wrote:If P is a prime number, then which of the following COULD be the sum of the factors of P?

1. 2P-1 2. P+1 3. P^2-1

A. 2 only
B. 2 and 3 only
C. 1 and 3 only
D. 1 only
E. 1,2 and 3

I think answer should be E.
Prime number only have 2 factors itself and 1. But as we have 2 as a prime number then we could have 1and 3 true as well.
2 has 2 factors 2 and 1. sum of the factors is 3. and hence all the 3 conditions are met. As it is a COULD be true questions so we can have answer E. If it were must be true then the answer will be A

Last edited by rakeshd347 on Sun Sep 29, 2013 11:45 pm, edited 1 time in total.

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faraz_jeddah: Master | Next Rank: 500 Posts; Posts: 358; Joined: Thu Apr 18, 2013 9:46 amFollowed by: 7 members; Location: Jeddah, Saudi Arabia; Thanked: 42 times; GMAT Score:730

by faraz_jeddah » Sun Sep 29, 2013 11:44 pm

vinay1983 wrote:If P is a prime number, then which of the following COULD be the sum of the factors of P?

1. 2P-1 2. P+1 3. P^2-1

A. 2 only
B. 2 and 3 only
C. 1 and 3 only
D. 1 only
E. 1,2 and 3

I will go with E

P + 1 is obvious

If P = 2 then sum of factors is 1+2 = 3
P^2 - 1 = 2^2 - 1 = 4 - 1 = 3

same goes for 2P - 1

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ganeshrkamath: Master | Next Rank: 500 Posts; Posts: 283; Joined: Sun Jun 23, 2013 11:56 pmFollowed by: 26 members; Location: Bangalore, India; Thanked: 97 times; GMAT Score:750

by ganeshrkamath » Mon Sep 30, 2013 12:22 am

vinay1983 wrote:If P is a prime number, then which of the following COULD be the sum of the factors of P?

1. 2P-1 2. P+1 3. P^2-1

A. 2 only
B. 2 and 3 only
C. 1 and 3 only
D. 1 only
E. 1,2 and 3

Take the prime number 2.
Sum of its factors = 1+2 = 3
2(2) - 1 = 3
2 + 1 = 3
2^2 - 1 = 3
Choose E

In general,
Sum of factors of a number = (a^(n+1) - 1)/(a-1) * (b^(m+1) - 1)/(b-1) * ...
where a,b,.. are the prime factors of the number.
and n,m,.. are their respective powers.

Example: 6 = 2*3
Sum of factors of 6 = (2^(1+1)-1)/(2-1) * (3^(1+1)-1)/(3-1)
= (3/1) * (8/2) = 12

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon Sep 30, 2013 5:58 am

vinay1983 wrote:If P is a prime number, then which of the following COULD be the sum of the factors of P?

1. 2P-1 2. P+1 3. P^2-1

A. 2 only
B. 2 and 3 only
C. 1 and 3 only
D. 1 only
E. 1,2 and 3

Picking numbers is a great way to go here.
Another option is to recognize that, if P is a prime number, then 1 and P are its ONLY factors.
So, the sum of the factors of P MUST EQUAL P + 1

Statement I: 2P - 1
Is it possible for the sum of the factors to be 2P - 1?
To find out, we'll set this equal to P + 1 to get . . .
2P - 1 = P + 1
Solve, to get P = 2
So, when P = 2, the sum of the factors is 2P - 1 (as well as P + 1)
Statement I can be true

Statement II: P + 1
P + 1 = P + 1, so statement II MUST be true

Statement III: PÂ² - 1
Set this equal to P + 1 to get . . .
PÂ² - 1 = P + 1
Rearrange: PÂ² - P - 2 = 0
Factor: (P - 2)(P + 1) = 0
Solve, to get P = 2 or P = -1 [-1 is not prime, so we won't consider it]
So, when P = 2, the sum of the factors is PÂ² - 1 (as well as P + 1)
Statement III can be true

Answer: E

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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vinay1983: Legendary Member; Posts: 643; Joined: Wed Aug 14, 2013 4:27 amFollowed by: 7 members; Thanked: 48 times

by vinay1983 » Mon Sep 30, 2013 6:10 am

So if i am right we are supposed to calculate/arrive at whether the statements "could be true" and not necessarily "must be true"

You can, for example never foretell what any one man will do, but you can say with precision what an average number will be up to!

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon Sep 30, 2013 6:11 am

vinay1983 wrote:So if i am right we are supposed to calculate/arrive at whether the statements "could be true" and not necessarily "must be true"

That's right.

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon Sep 30, 2013 6:24 am

Notice that, if we set each answer choice equal to P + 1 (as I did in my solution above), we can handle much trickier statements.
For example, COULD PÂ² - 4P - 13 be the sum of the factors of P?
What about PÂ² - 7P - 15?

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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by [email protected] » Mon Sep 30, 2013 11:16 pm

Hi vinay1983,

What is the source for this question? I only ask because it doesn't follow the typical GMAT style for Roman Numeral questions (which would ask "what MUST be true?"), it can be beaten by using the easiest number possible (the number 2, without testing the thoroughness of your knowledge) and doesn't have any of the "frequency shortcuts" that are typical of Roman Numeral questions. Roman Numeral questions on the GMAT are usually a bit more "involved" than this question.

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