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Difference between revisions of "2017 AMC 10B Problems/Problem 23"

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==Problem 23==
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==Problem==
 
Let <math>N=123456789101112\dots4344</math> be the <math>79</math>-digit number that is formed by writing the integers from <math>1</math> to <math>44</math> in order, one after the other. What is the remainder when <math>N</math> is divided by <math>45</math>?
 
Let <math>N=123456789101112\dots4344</math> be the <math>79</math>-digit number that is formed by writing the integers from <math>1</math> to <math>44</math> in order, one after the other. What is the remainder when <math>N</math> is divided by <math>45</math>?
  
 
<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>
  
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==Solution==
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==Solution 1==
 
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>.
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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. Therefore, by inspection, the answer is <math>\boxed{\textbf{(C) } 9}</math>.
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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 the [[Chinese Remainder Theorem]] we get the remainder to be <math>9 \pmod{45}</math>. Therefore, the answer is <math>\boxed{\textbf{(C) } 9}</math>.
  
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Note: the sum of the digits of <math>N</math> is <math>270</math>.
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===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 <math>45 \equiv 4 \text{ (mod 5)}</math>.
  
 
==Solution 2==
 
==Solution 2==
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Noting the solution above, we try to find the sum of the digits to figure out its remainder when divided by <math>9</math>. From <math>1</math> thru <math>9</math>, the sum is <math>45</math>. <math>10</math> thru <math>19</math>, the sum is <math>55</math>, <math>20</math> thru <math>29</math> is <math>65</math>, and <math>30</math> thru <math>39</math> is <math>75</math>. Thus the sum of the digits is <math>45+55+65+75+4+5+6+7+8 = 240+30 = 270</math>, and thus <math>N</math> is divisible by <math>9</math>. Now, refer to the above solution. <math>N \equiv 4 \text{ (mod 5)}</math> and <math>N \equiv 0 \text{ (mod 9)}</math>. From this information, we can conclude that <math>N \equiv 54 \text{ (mod 5)}</math> and <math>N \equiv 54 \text{ (mod 9)}</math>. Therefore, <math>N \equiv 54 \text{ (mod 45)}</math> and <math>N \equiv 9 \text{ (mod 45)}</math> so the remainder is <math>\boxed{\textbf{(C) }9}</math>
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Realize that <math>10 \equiv 10 \cdot 10 \equiv 10^{k} \pmod{45}</math> for all positive integers <math>k</math>.
  
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Apply this on the expanded form of <math>N</math>:
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<cmath>N = 1(10)^{78} + 2(10)^{77} + \cdots + 9(10)^{70} + 10(10)^{68} + 11(10)^{66} + \cdots + 43(10)^{2} + 44 \equiv</cmath>
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<cmath>10(1 + 2 + \cdots + 43) + 44 \equiv 10 \left (\frac{43 \cdot 44}2 \right ) + 44 \equiv</cmath>
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<cmath>10 \left (\frac{-2 \cdot -1}2 \right ) - 1 \equiv \boxed{\textbf{(C) } 9} \pmod{45}</cmath>
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==Solution 3 (Clever way using divisibility rules)==
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We know that <math>45 = 5 \cdot 9</math>, so we can apply our restrictions to that. We know that the units digit must be <math>5</math> or <math>0</math>, and the digits must add up to a multiple of <math>9</math>. <math>1+2+3+4+\cdots + 44 = \frac{44 \cdot 45}{2}</math>. We can quickly see this is a multiple of <math>9</math> because <math>\frac{44}{2} \cdot 45 = 22 \cdot 45</math>. We know <math>123 \ldots 4344</math> is not a multiple of <math>5</math> because the units digit isn't <math>5</math> or <math>0</math>. We can just subtract by 9 until we get a number whose units digit is 5 or 0.
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We have <math>123 \ldots 4344</math> is divisible by <math>9</math>, so we can subtract by <math>9</math> to get <math>123 \ldots 4335</math> and we know that this is divisible by 5. So our answer is <math>\boxed{\textbf{(C) } 9}</math>
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 +
~Arcticturn
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==Solution 4==
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Since <math>N</math> ends with the number <math>4</math>, <math>N\equiv 4\pmod 5</math>. Also, the sum of the digits of <math>N</math> are divisible by <math>9</math>, so <math>N</math> must be divisible by <math>9</math>. Therefore, we have the system of equations:
 +
<math>N\equiv 4\pmod 5</math>
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<math>N\equiv 0\pmod 9</math>
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According to the second equation, <math>N\equiv \{0, 9, 18, 27, 36\} \pmod {45}</math>. The only one of these solutions that is <math>4\pmod 5</math> is <math>\boxed{\textbf{(C) } 9}</math>
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~[https://artofproblemsolving.com/wiki/index.php/User:Sid2012 sid2012]
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2017|ab=B|num-b=22|num-a=24}}
 
{{AMC10 box|year=2017|ab=B|num-b=22|num-a=24}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 19:25, 4 November 2024

Problem

Let $N=123456789101112\dots4344$ be the $79$-digit number that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 44$

Solution 1

We only need to find the remainders of N when divided by 5 and 9 to determine the answer. By inspection, $N \equiv 4 \text{ (mod 5)}$. The remainder when $N$ is divided by $9$ is $1+2+3+4+ \cdots +1+0+1+1 +1+2 +\cdots + 4+3+4+4$, but since $10 \equiv 1 \text{ (mod 9)}$, we can also write this as $1+2+3 +\cdots +10+11+12+ \cdots 43 + 44 = \frac{44 \cdot 45}2 = 22 \cdot 45$, which has a remainder of 0 mod 9. Solving these modular congruence using the Chinese Remainder Theorem we get the remainder to be $9 \pmod{45}$. Therefore, the answer is $\boxed{\textbf{(C) } 9}$.

Alternative Ending to Solution 1

Once we find our 2 modular congruences, we can narrow our options down to ${C}$ and ${D}$ because the remainder when $N$ is divided by $45$ should be a multiple of 9 by our modular congruence that states $N$ has a remainder of $0$ when divided by $9$. Also, our other modular congruence states that the remainder when divided by $45$ should have a remainder of $4$ when divided by $5$. Out of options $C$ and $D$, only $\boxed{\textbf{(C) } 9}$ satisfies that the remainder when $N$ is divided by $45 \equiv 4 \text{ (mod 5)}$.

Solution 2

Realize that $10 \equiv 10 \cdot 10 \equiv 10^{k} \pmod{45}$ for all positive integers $k$.

Apply this on the expanded form of $N$: \[N = 1(10)^{78} + 2(10)^{77} + \cdots + 9(10)^{70} + 10(10)^{68} + 11(10)^{66} + \cdots + 43(10)^{2} + 44 \equiv\] \[10(1 + 2 + \cdots + 43) + 44 \equiv 10 \left (\frac{43 \cdot 44}2 \right ) + 44 \equiv\] \[10 \left (\frac{-2 \cdot -1}2 \right ) - 1 \equiv \boxed{\textbf{(C) } 9} \pmod{45}\]

Solution 3 (Clever way using divisibility rules)

We know that $45 = 5 \cdot 9$, so we can apply our restrictions to that. We know that the units digit must be $5$ or $0$, and the digits must add up to a multiple of $9$. $1+2+3+4+\cdots + 44 = \frac{44 \cdot 45}{2}$. We can quickly see this is a multiple of $9$ because $\frac{44}{2} \cdot 45 = 22 \cdot 45$. We know $123 \ldots 4344$ is not a multiple of $5$ because the units digit isn't $5$ or $0$. We can just subtract by 9 until we get a number whose units digit is 5 or 0.

We have $123 \ldots 4344$ is divisible by $9$, so we can subtract by $9$ to get $123 \ldots 4335$ and we know that this is divisible by 5. So our answer is $\boxed{\textbf{(C) } 9}$

~Arcticturn

Solution 4

Since $N$ ends with the number $4$, $N\equiv 4\pmod 5$. Also, the sum of the digits of $N$ are divisible by $9$, so $N$ must be divisible by $9$. Therefore, we have the system of equations: 

$N\equiv 4\pmod 5$
$N\equiv 0\pmod 9$

According to the second equation, $N\equiv \{0, 9, 18, 27, 36\} \pmod {45}$. The only one of these solutions that is $4\pmod 5$ is $\boxed{\textbf{(C) } 9}$

~sid2012

See Also

2017 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
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All AMC 10 Problems and Solutions

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