Difference between revisions of "2017 AMC 10B Problems/Problem 23"
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==Alternative Ending to Solution 1== | ==Alternative Ending to Solution 1== | ||
Once we find our 2 modular congruences, we can narrow our options down to <math>{C}</math> and <math>{D}</math> because the remainder when <math>N</math> is divided by <math>45</math> should be a multiple of 9 by our modular congruence that states <math>N</math> has a remainder of <math>0</math> when divided by <math>9</math>. Also, our other modular congruence states that the remainder when divided by <math>45</math> should have a remainder of <math>4</math> when divided by <math>5</math>. Out of options <math>C</math> and <math>D</math>, only <math>\boxed{\textbf{(C) } 9}</math> satisfies that the remainder when <math>N</math> is divided by 45 <math>\equiv 4 \text{ (mod 5)}</math>. | Once we find our 2 modular congruences, we can narrow our options down to <math>{C}</math> and <math>{D}</math> because the remainder when <math>N</math> is divided by <math>45</math> should be a multiple of 9 by our modular congruence that states <math>N</math> has a remainder of <math>0</math> when divided by <math>9</math>. Also, our other modular congruence states that the remainder when divided by <math>45</math> should have a remainder of <math>4</math> when divided by <math>5</math>. Out of options <math>C</math> and <math>D</math>, only <math>\boxed{\textbf{(C) } 9}</math> satisfies that the remainder when <math>N</math> is divided by 45 <math>\equiv 4 \text{ (mod 5)}</math>. | ||
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==Solution 3== | ==Solution 3== |
Revision as of 14:09, 27 September 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 3
Realize that for all positive integers
.
Apply this on the expanded form of :
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 |
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