Difference between revisions of "2014 AMC 12B Problems/Problem 8"
Kevin38017 (talk | contribs) (Created page with "==Problem== In the addition shown below <math> A </math>, <math> B </math>, <math> C </math>, and <math> D </math> are distinct digits. How many different values are possible fo...") |
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From the first column, we see <math>A+B < 10</math> because it yields a single digit answer. From the fourth column, we see that <math>C+D</math> equals <math>D</math> and therefore <math>C = 0</math>. We know that <math>A+B = D</math>. Therefore, the number of values <math>D</math> can take is equal to the number of possible sums less than <math>10</math> that can be formed by adding two distinct natural numbers. Letting <math>A=1</math>, and letting <math>B=2,3,4,5,6,7,8</math>, we have | From the first column, we see <math>A+B < 10</math> because it yields a single digit answer. From the fourth column, we see that <math>C+D</math> equals <math>D</math> and therefore <math>C = 0</math>. We know that <math>A+B = D</math>. Therefore, the number of values <math>D</math> can take is equal to the number of possible sums less than <math>10</math> that can be formed by adding two distinct natural numbers. Letting <math>A=1</math>, and letting <math>B=2,3,4,5,6,7,8</math>, we have | ||
<cmath>D = 3,4,5,6,7,8,9 \implies \boxed{\textbf{(C)}\ 7}</cmath> | <cmath>D = 3,4,5,6,7,8,9 \implies \boxed{\textbf{(C)}\ 7}</cmath> | ||
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Revision as of 22:36, 20 February 2014
Problem
In the addition shown below ,
,
, and
are distinct digits. How many different values are possible for
?
Solution
From the first column, we see because it yields a single digit answer. From the fourth column, we see that
equals
and therefore
. We know that
. Therefore, the number of values
can take is equal to the number of possible sums less than
that can be formed by adding two distinct natural numbers. Letting
, and letting
, we have