Difference between revisions of "2015 AMC 10B Problems/Problem 10"

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Therefore, the answer must be <math>\boxed{\text{(\textbf C) It is a negative number ending with a 5.}}</math>
 
Therefore, the answer must be <math>\boxed{\text{(\textbf C) It is a negative number ending with a 5.}}</math>
  
Alternatively, we can calculate the units digit of each group of 5 odd numbers using modular arithmetic
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Alternatively, we can calculate the units digit of each group of 5 odd numbers using modular arithmetic.
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 +
==Video Solution 1==
 +
https://youtu.be/r5QASm5hnII
 +
 
 +
~Education, the Study of Everything
 +
 
 +
==Video Solution==
 +
https://youtu.be/UoU-cvet1pg
 +
 
 +
~savannahsolver
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2015|ab=B|num-b=9|num-a=11}}
 
{{AMC10 box|year=2015|ab=B|num-b=9|num-a=11}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 17:13, 2 August 2022

Problem

What are the sign and units digit of the product of all the odd negative integers strictly greater than $-2015$?
$\textbf{(A) }$ It is a negative number ending with a 1.
$\textbf{(B) }$ It is a positive number ending with a 1.
$\textbf{(C) }$ It is a negative number ending with a 5.
$\textbf{(D) }$ It is a positive number ending with a 5.
$\textbf{(E) }$ It is a negative number ending with a 0.

Solution

Since $-5>-2015$, the product must end with a $5$.

The multiplicands are the odd negative integers from $-1$ to $-2013$. There are $\frac{|-2013+1|}2+1=1006+1$ of these numbers. Since $(-1)^{1007}=-1$, the product is negative.

Therefore, the answer must be $\boxed{\text{(\textbf C) It is a negative number ending with a 5.}}$

Alternatively, we can calculate the units digit of each group of 5 odd numbers using modular arithmetic.

Video Solution 1

https://youtu.be/r5QASm5hnII

~Education, the Study of Everything

Video Solution

https://youtu.be/UoU-cvet1pg

~savannahsolver

See Also

2015 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
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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