Difference between revisions of "2017 AMC 10A Problems/Problem 20"
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==Solution== | ==Solution== | ||
− | Note that <math>n \equiv S(n) \pmod{9}</math>. This can be seen from the fact that <math>\sum_{k=0}^{n}10^{k}a_k \equiv \sum_{k=0}^{n}a_k \pmod{9}</math>. Thus, if <math>S(n) = 1274</math>, then <math>n \equiv 5 \pmod{9}</math>, and thus <math>n+1 \equiv S(n+1) \equiv 6 \pmod{9}</math>. The only answer choice that is <math>6 \pmod{9}</math> is <math>\boxed{\textbf{(D)}\ | + | Note that <math>n \equiv S(n) \pmod{9}</math>. This can be seen from the fact that <math>\sum_{k=0}^{n}10^{k}a_k \equiv \sum_{k=0}^{n}a_k \pmod{9}</math>. Thus, if <math>S(n) = 1274</math>, then <math>n \equiv 5 \pmod{9}</math>, and thus <math>n+1 \equiv S(n+1) \equiv 6 \pmod{9}</math>. The only answer choice that is <math>6 \pmod{9}</math> is <math>\boxed{\textbf{(D)}\ 1239}</math>. |
==See Also== | ==See Also== | ||
{{AMC10 box|year=2017|ab=A|num-b=19|num-a=21}} | {{AMC10 box|year=2017|ab=A|num-b=19|num-a=21}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 18:29, 8 February 2017
Problem
Let equal the sum of the digits of positive integer . For example, . For a particular positive integer , . Which of the following could be the value of ?
Solution
Note that . This can be seen from the fact that . Thus, if , then , and thus . The only answer choice that is is .
See Also
2017 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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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