For how many positive integers x does 1250x have fewer

Question

For how many positive integers  x  does  1250x  have fewer than 20 positive divisors?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

paragw · Answer

Here's my solution:

MarkGMATMentor · Answer

This question hinges on knowing how to find the number of positive divisors (or factors) of a positive integer using the    prime factor    method .

First find the prime factors:
 1250  -->  10 * 125  -->  2 * 5 * 5^3  -->  2^1 * 5^4

Then, add 1 to the exponent of each prime and multiply the results to get the total number of positive divisors:
 (1+1) * (4+1) = 2*5 = 10

So, 1250 already has 10 positive divisors. If we were to make x a prime other than 2 or 5 (say 3), then it would have an exponent of 1. The prime factorization would then be  2^1 * 5^4*3^1 . Adding 1 to the exponents would then result in multiplying 10 by 2, giving us 20 positive divisors. So, x cannot contain any "new" primes. What can it be?

First, x could simply be 1, which wouldn't change the number of divisors.

The only other possibilities are more 2's, more 5's, or both. Let's make a table:
   x 
 prime factors of 1250x 
 divisors of 1250x
(add 1 to exponents
and multiply) 
 
  1 
  2^1*5^4  
 2*5 = 10 
 
  2 
  2^2*5^4  
 3*5 = 15 
 
  5 
  2^1*5^5  
 2*6 = 12 
 
   5^2  
  2^1*5^6  
 2*7 = 14 
 
   5^3  
  2^1*5^7  
 2*8 = 16 
 
   5^4  
  2^1*5^8  
 2*9 = 18 
 
  2*5 
  2^2*5^5  
 3*6 = 18

Any other values for x will result in 20 or more positive divisors. So, there are 7 possibilities for x and the answer is E.

Make sure you understand the prime factor method for finding the number of positive divisors of an integer. If you get a question like this, the prime factor method is essential to solve within a reasonable amount of time.

For how many positive integers x does 1250x have fewer

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x	prime factors of 1250x	divisors of 1250x (add 1 to exponents and multiply)
1	\(2^1*5^4\)	2*5 = 10
2	\(2^2*5^4\)	3*5 = 15
5	\(2^1*5^5\)	2*6 = 12
\(5^2\)	\(2^1*5^6\)	2*7 = 14
\(5^3\)	\(2^1*5^7\)	2*8 = 16
\(5^4\)	\(2^1*5^8\)	2*9 = 18
2*5	\(2^2*5^5\)	3*6 = 18

GMAT Problem Solving (PS) Questions

For how many positive integers x does 1250x have fewer

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