Difference between revisions of "2014 AMC 10B Problems/Problem 10"
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==Solution== | ==Solution== | ||
+ | Note from the addition of the last digits that <math>A+B=D\text{ or }D+10</math>. In the latter case we must have that <math>1+C+D=D\text{ or }10+D</math>, implying that <math>C=9</math>. In the addition of the third digits, we then have <math>1+B+A=D\text{ or }D+10</math>, a contradiction from our assumption that <math>A+B=10</math>. Thus <math>A+B=D</math>. | ||
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+ | This then implies that <math>C+D=D</math>, or <math>C=0</math>. Note that all of the remaining equalities are now satisfied: <math>A+B=D, B+C=B,</math> and <math>B+A=D</math>. Thus, if we have some <math>A,B,D</math> such that <math>A+B=D</math> then the addition will be satisfied. Since the digits must be distinct, the smallest possible value of <math>D</math> is <math>1+2=3</math>, and the largest possible value is <math>9</math>. Any of these values can be obtained by taking <math>A=1,B=D-1</math>. Thus we have that <math>3\le D\le9</math>, so the number of possible values is <math>\boxed{\textbf{(C) }7</math> | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2014|ab=B|num-b=9|num-a=11}} | {{AMC10 box|year=2014|ab=B|num-b=9|num-a=11}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 18:35, 20 February 2014
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 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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