Determine the maximum value of a prime number x ≤ 999, such that Y has precisely 42 distinct positive integer divisors (including 1 and Y), where:
Y = x(x+1)².

x(x+1)˛ has 42 divisors.

x is relatively prime to (x+1)

=>(x+1)˛ has 21(=3*7) divisors

x+1 = pł * q (p,q are primes)

For x to be prime, x+1 has to be even => 

p=2 or q=2

Case(1): q=2

p=3 or 5 or 7 => x = 53 or 249(no prime) or 685(no prime)

Case(2): p=2

Find prime q such that 8q-1 is prime

Through spreadsheet: q=103 x=823 satisfy the solution

=> x =823 is the max prime number

Posted by Praneeth on 2010-06-18 08:15:17