Question

How many positive integers can fill the blank in the sentence below?

“One positive integer is ___ more than twice another, and the sum of the two numbers is $28$ ”

$6$

$7$

$8$

$9$

$10$

Let the smaller number be $x$ , and let the blank be $c$ . Then the other number is $2x+c$ , where both $x$ and $c$ are positive integers.

The sum condition gives $x+(2x+c)=28$ , so $c=28-3x$ . For $c$ to be positive, 28-3x>0, so $x<\frac{28}{3}$ .

Thus $x$ can be any integer from $1$ through $9$ , giving $9$ possible values of the blank.

Thus, the correct answer is D.