2016 AMC 10B Problems/Problem 6

Problem

Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number $S$ . What is the smallest possible value for the sum of the digits of $S$ ?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20$

Solution

Since we know that the sum of two three digit positive integers cannot start with a one, we can get rid of option A. Since $4$ is the next smallest integer, we can try to make an integer that starts with $4$ and ends with $00$ We can find that $143+257$ fits the parameters(along with many others), in which all the digits are distinct, and indeed sum to $400$ , summing the digits we reach the correct answer choice; $\textbf{(B)}\ 4$ .

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2016 AMC 10B Problems/Problem 6

Problem

Solution