Difference between revisions of "2017 AMC 10B Problems/Problem 23"
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<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 44</math> | <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 44</math> | ||
| − | + | ==Solution== | |
| − | + | We only need to find the remainders of N when divided by 5 and 9 to determine the answer. | |
| + | By inspection, <math>N \equiv 4 \text{ (mod 5)}</math>. | ||
| + | The remainder when divided by 9 is <math>1+2+3+4 \cdot 1+0+1+1 \cdot 4+3+4+4</math>, but since <math>10 \equiv 1 \text{ (mod 9)}</math>, we can also write this as <math>1+2+3 \cdot 10+11+12 \cdot 43 + 44 = \frac{44 \cdot 45}2 = 22 \cdot 45</math>, which clearly has a remainder of 0 mod 9. Therefore, by CRT, the answer is <math>\boxed{\textbf{(C) } 9}</math>. | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2017|ab=B|num-b=24|after=Last Problem}} | {{AMC10 box|year=2017|ab=B|num-b=24|after=Last Problem}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 09:43, 16 February 2017
Problem 23
Let 
be the 
-digit number that is formed by writing the integers from 
to 
in order, one after the other. What is the remainder when 
is divided by 
?
Solution
We only need to find the remainders of N when divided by 5 and 9 to determine the answer.
By inspection, 
.
The remainder when divided by 9 is 
, but since 
, we can also write this as 
, which clearly has a remainder of 0 mod 9. Therefore, by CRT, the answer is 
.
See Also
| 2017 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 24 |
Followed by Last Problem | |
| 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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