2014 AMC 10B Problems/Problem 10
Problem
In the addition shown below ,
,
, and
are distinct digits. How many different values are possible for
?
Solution
Note from the addition of the last digits that . In the latter case we must have that
, implying that
. In the addition of the third digits, we then have
, a contradiction from our assumption that
. Thus
.
This then implies that , or
. Note that all of the remaining equalities are now satisfied:
and
. Thus, if we have some
such that
then the addition will be satisfied. Since the digits must be distinct, the smallest possible value of
is
, and the largest possible value is
. Any of these values can be obtained by taking
. Thus we have that
, so the number of possible values is $\boxed{\textbf{(C) }7$ (Error compiling LaTeX. Unknown error_msg)
See Also
2014 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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All AMC 10 Problems and Solutions |
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