AMC8 2022

AMC8 2022 · Q13

AMC8 2022 · Q13. It mainly tests Linear equations, Linear inequalities.

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 $28$.”

下面句子中的空白可以填入多少个正整数？ “一个正整数比另一个正整数多_____，并且两个数的和是28。”

(A) 6 6

(B) 7 7

(D) 9 9

(E) 10 10

Answer

Correct choice: (D)

正确答案：（D）

Solution

Let $m$ and $n$ be positive integers such that $m>n$ and $m+n=28.$ It follows that $m=2n+d$ for some positive integer $d.$ We wish to find the number of possible values for $d.$ By substitution, we have $(2n+d)+n=28,$ from which $d=28-3n.$ Note that $n=1,2,3,\ldots,9$ each generate a positive integer for $d,$ so there are $\boxed{\textbf{(D) } 9}$ possible values for $d.$

设$m$和$n$为正整数，$m>n$且$m+n=28$。则$m=2n+d$，其中$d$为正整数。由代入，得$(2n+d)+n=28$，即$d=28-3n$。$n=1,2,3,\ldots,9$时$d$均为正整数，故有$\boxed{\textbf{(D) } 9}$个可能值。

Topics