What are two numbers whose greatest common factor is 7 and whose sum is 105 such that the larger is twice the smaller and the smaller has only two prime factors?

Since one number is twice the other, the smaller number must be the greatest common factor. Since the greatest common factor is 7, that would make the other number 14. But, 7 is a Prime number and has only one prime factor. However, the larger number, 14, has two prime factors. Also, the sum of the two numbers is 21, not 105. So, the information in the problem does not have a solution.

Let us ignore the greatest common factor information. Let the smaller number be x. That means the larger number is 2x.

x + 2x = 105 => 3x = 105 => x = 35.

The two numbers are 35 and 70. The greatest common factor is 35. The smaller number, 35, has only two prime factors.