Aman verma wrote:Q:Find all the primes P such that the sum of all positive integral divisors of P^4 is equal to square of an integer.[ P^4= P raised to the power 4.]
A) 3,5,7,17
B) 5
C) 7,13,19,23,29,31,61
D) 3
E) Infinitely many
Yikes, I have no idea how to solve this question!
Unless I'm missing something totally basic (which is quite possible), I believe that this question requires math that is beyond the scope of the GMAT.
Having said that, I'll share with you how I'd tackle this very difficult (for me!) question on test day.
First, check the answer choices. Notice that 3 is a possible P value for some answer choices bt ot others. So, let's check P = 3
If P = 3, then P^4 = 81
Sum of divisors of 81 = 1+3+9+27+81 = 121 = 11². So, 3 is one possible value of P
Since answer choices B and C don't include 3 as a possible value of P, we can ELIMINATE them.
So, the correct answer is A, D or E.
Since answer choice A includes 5 as a possible P value, let's check P = 5
If P = 5, then P^4 = 625
Sum of divisors of 625 = 1+5+25+125+625 = 781. Since 781 does not equal the square of an integer, 5 is NOT a possible value of P
So, we can ELIMINATE A.
This leaves us with answer choices D or E.
At this point, I might check 2 as a possible P value, since 2^4 is a relatively small number to work with. If P = 2 works, then I can eliminate D.
If P = 2, then P^4 = 16
Sum of divisors of 16 = 1+2+4+8+16 = 31. Since 31 does not equal the square of an integer, 2 is NOT a possible value of P
So, we
cannot eliminate D.
At this point, we should recognize that testing other possible primes will take FAR TOO MUCH TIME.
So, I'd guess either D or E (my guess is
D) and move onto a (hopefully) easier question.
Cheers,
Brent
