Difference between revisions of "2014 AMC 10A Problems/Problem 20"

(added formal proof to solution 1 for those who would like a more rigorous version of it.)
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<math>\textbf{(A)}\ 901\qquad\textbf{(B)}\ 911\qquad\textbf{(C)}\ 919\qquad\textbf{(D)}\ 991\qquad\textbf{(E)}\ 999</math>
 
<math>\textbf{(A)}\ 901\qquad\textbf{(B)}\ 911\qquad\textbf{(C)}\ 919\qquad\textbf{(D)}\ 991\qquad\textbf{(E)}\ 999</math>
  
==Solution==
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==Solution 1==
 
We can list the first few numbers in the form <math>8 \cdot (8....8)</math>
 
We can list the first few numbers in the form <math>8 \cdot (8....8)</math>
  
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<math>8 \cdot 88888 = 711104</math>
 
<math>8 \cdot 88888 = 711104</math>
  
By now it's clear that the numbers will be in the form <math>7</math>, <math>k-2</math> <math>1</math>s, and <math>04</math>. We want to make the numbers sum to 1000, so <math>7+4+(k-2) = 1000</math>. Solving, we get <math>k = 991</math>, meaning the answer is <math>\fbox{(D)}</math>
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By now it's clear that the numbers will be in the form <math>7</math>, <math>k-2</math> <math>1</math>'s, and <math>04</math>. We want to make the numbers sum to 1000, so <math>7+4+(k-2) = 1000</math>. Solving, we get <math>k = 991</math>, meaning the answer is <math>\fbox{(D)}</math>
  
Another way to proceed is that we know the difference between the sum of the digits of each product and <math>k</math> is always <math>9</math>, so we just do <math>1000-9=\boxed{\textbf{(D)991}}</math>.
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Another way to proceed is that we know the difference between the sum of the digits of each product and <math>k</math> is always <math>9</math>, so we just do <math>1000-9=\boxed{\textbf{(D) } 991}</math>.
  
===Proof of this solution's validity===
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===Proof of Solution 1===
  
 
Since this solution won't fly on a proof-based competition, here's a proof that it's valid:
 
Since this solution won't fly on a proof-based competition, here's a proof that it's valid:
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<cmath>x_{k+1}=64\times 10^{k+1}+x_k</cmath>
 
<cmath>x_{k+1}=64\times 10^{k+1}+x_k</cmath>
  
We will now use induction, Our base case will be <math>k=2</math>. It's easy to see that this becomes <math>x_2=704</math>. Then, the <math>k+1</math> case: let x_k=7111\dots104 with <math>k-2</math> <math>1</math>s. Then <math>x_{k+1}=64000\dots000+7111\dots104</math>. Adding these numbers, we get <math>x_{k+1}=71111\dots104</math>.  
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We will now use induction, Our base case will be <math>k=2</math>. It's easy to see that this becomes <math>x_2=704</math>. Then, the <math>k+1</math> case: let <math>x_k=7111\dots104</math> with <math>k-2</math> <math>1</math>s. Then <math>x_{k+1}=64000\dots000+7111\dots104</math>. Adding these numbers, we get <math>x_{k+1}=71111\dots104</math>.  
  
 
Summing these digits, we have <math>4+7+(k-2)=1000</math>, giving us <math>k=991</math>.
 
Summing these digits, we have <math>4+7+(k-2)=1000</math>, giving us <math>k=991</math>.
  
==Solution 2(Educated Guesses if you have no time)==
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==Solution 2 (Educated Guessing if you have no time)==
 
We first note that <math>125 \cdot 8 = 1000</math> and so we assume there are <math>125</math> 8s.
 
We first note that <math>125 \cdot 8 = 1000</math> and so we assume there are <math>125</math> 8s.
  
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Thus we make an educated guess that it is somehow less by 1, so we get <math>\fbox{(D)}</math>. ~mathboy282
 
Thus we make an educated guess that it is somehow less by 1, so we get <math>\fbox{(D)}</math>. ~mathboy282
  
===Note(Must Read)===
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===Note (Must Read)===
We were just lucky; this method is NOT reliable. Please note that this probably will not work for other problems and is just a lucky coincidence.
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We were just lucky; this method is NOT reliable. Please note that this probably will not work for other problems; it is just a lucky coincidence.
  
 
==Video Solution==
 
==Video Solution==

Latest revision as of 12:19, 7 September 2021

The following problem is from both the 2014 AMC 12A #16 and 2014 AMC 10A #20, so both problems redirect to this page.

Problem

The product $(8)(888\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is $k$?

$\textbf{(A)}\ 901\qquad\textbf{(B)}\ 911\qquad\textbf{(C)}\ 919\qquad\textbf{(D)}\ 991\qquad\textbf{(E)}\ 999$

Solution 1

We can list the first few numbers in the form $8 \cdot (8....8)$

(Hard problem to do without the multiplication, but you can see the pattern early on)

$8 \cdot 8 = 64$

$8 \cdot 88 = 704$

$8 \cdot 888 = 7104$

$8 \cdot 8888 = 71104$

$8 \cdot 88888 = 711104$

By now it's clear that the numbers will be in the form $7$, $k-2$ $1$'s, and $04$. We want to make the numbers sum to 1000, so $7+4+(k-2) = 1000$. Solving, we get $k = 991$, meaning the answer is $\fbox{(D)}$

Another way to proceed is that we know the difference between the sum of the digits of each product and $k$ is always $9$, so we just do $1000-9=\boxed{\textbf{(D) } 991}$.

Proof of Solution 1

Since this solution won't fly on a proof-based competition, here's a proof that it's valid:

We will call $x_k=8(888\dots8)$ with exactly $k$ $8$s. We then rewrite this more formally, as:

\[x_k=8\biggr(\sum_{n=0}^{k}8(10)^n\biggr)\] \[=64\biggr(\sum_{n=0}^{k}(10)^n\biggr)\] \[=64\frac{10^{k+1}-1}{9}\]

Then, finding a recursive formula, we get:

\[x_{k+1}=64\times 10^{k+1}+x_k\]

We will now use induction, Our base case will be $k=2$. It's easy to see that this becomes $x_2=704$. Then, the $k+1$ case: let $x_k=7111\dots104$ with $k-2$ $1$s. Then $x_{k+1}=64000\dots000+7111\dots104$. Adding these numbers, we get $x_{k+1}=71111\dots104$.

Summing these digits, we have $4+7+(k-2)=1000$, giving us $k=991$.

Solution 2 (Educated Guessing if you have no time)

We first note that $125 \cdot 8 = 1000$ and so we assume there are $125$ 8s.

Then we note that it is asking for the second factor, so we subtract $1$(the original $8$ in the first factor).

Now we have $125-1=124.$ The second factor is obviously a multiple of $124$.

Listing the first few, we have $124, 248, 372, 496, 620, 744, 868, 992, 1116, 1240, ...$

We notice that the 4th answer choice is 1 less than a 992(a multiple of 124.)

Thus we make an educated guess that it is somehow less by 1, so we get $\fbox{(D)}$. ~mathboy282

Note (Must Read)

We were just lucky; this method is NOT reliable. Please note that this probably will not work for other problems; it is just a lucky coincidence.

Video Solution

https://youtu.be/wQzuQZvq8sk

~savannahsolver

See Also

2014 AMC 10A (ProblemsAnswer KeyResources)
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
2014 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 15
Followed by
Problem 17
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 12 Problems and Solutions

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