Difference between revisions of "2017 AMC 10B Problems/Problem 23"
Stanleyh0608 (talk | contribs) m (→Solution2) |
m (→Solution 1) |
||
| Line 8: | Line 8: | ||
We only need to find the remainders of N when divided by 5 and 9 to determine the answer. | 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>. | By inspection, <math>N \equiv 4 \text{ (mod 5)}</math>. | ||
| − | The remainder when <math>N</math> is divided by <math>9</math> is <math>1+2+3+4+ \cdots +1+0+1+1 +1+2 +\cdots + 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 +\cdots +10+11+12+ \cdots 43 + 44 = \frac{44 \cdot 45}2 = 22 \cdot 45</math>, which has a remainder of 0 mod 9. Solving these modular congruence using CRT(Chinese Remainder Theorem) we | + | The remainder when <math>N</math> is divided by <math>9</math> is <math>1+2+3+4+ \cdots +1+0+1+1 +1+2 +\cdots + 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 +\cdots +10+11+12+ \cdots 43 + 44 = \frac{44 \cdot 45}2 = 22 \cdot 45</math>, which has a remainder of 0 mod 9. Solving these modular congruence using CRT(Chinese Remainder Theorem) we get the remainder to be <math>9 \pmod{45}</math>. Therefore, the answer is <math>\boxed{\textbf{(C) } 9}</math>. |
==Alternative Ending to Solution 1== | ==Alternative Ending to Solution 1== | ||
Revision as of 20:56, 26 June 2020
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 1
We only need to find the remainders of N when divided by 5 and 9 to determine the answer.
By inspection, 
.
The remainder when is divided by 
is 
, but since 
, we can also write this as 
, which has a remainder of 0 mod 9. Solving these modular congruence using CRT(Chinese Remainder Theorem) we get the remainder to be 
. Therefore, the answer is 
.
Alternative Ending to Solution 1
Once we find our 2 modular congruences, we can narrow our options down to 
and 
because the remainder when is divided by should be a multiple of 9 by our modular congruence that states has a remainder of 
when divided by . Also, our other modular congruence states that the remainder when divided by should have a remainder of 
when divided by 
. Out of options 
and 
, only satisfies that the remainder when is divided by 45 
.
Solution 2
The same way, you can get N==4(Mod 5) and 0(Mod 9). By The Chinese remainder Theorem, the answer comes out to be 
See Also
| 2017 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
