AMC12 2021 A

AMC12 2021 A · Q3

AMC12 2021 A · Q3. It mainly tests Fractions, GCD & LCM.

The sum of two natural numbers is $17{,}402$. One of the two numbers is divisible by $10$. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?

两个自然数的和是$17{,}402$。其中一个数能被$10$整除。如果擦除该数的个位数字，就得到另一个数。这两个数的差是多少？

(A) 10{,}272 10{,}272

(B) 11{,}700 11{,}700

(D) 14{,}238 14{,}238

(E) 15{,}426 15{,}426

Answer

Correct choice: (D)

正确答案：（D）

Solution

The units digit of a multiple of $10$ will always be $0$. We add a $0$ whenever we multiply by $10$. So, removing the units digit is equal to dividing by $10$. Let the smaller number (the one we get after removing the units digit) be $a$. This means the bigger number would be $10a$. We know the sum is $10a+a = 11a$ so $11a=17402$. So $a=1582$. The difference is $10a-a = 9a$. So, the answer is $9(1582) = \boxed{\textbf{(D)} ~14{,}238}$.

$10$的倍数的个位数字总是$0$。我们每次乘以$10$就是加一个$0$。所以，擦除个位数字等于除以$10$。设较小的数（擦除个位数字后得到的数）为$a$。这意味着较大的数是$10a$。我们知道和是$10a+a = 11a$，所以$11a=17402$。于是$a=1582$。差是$10a-a = 9a$。所以，答案是$9(1582) = \boxed{\textbf{(D)} ~14{,}238}$。

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