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Difference between revisions of "2020 AMC 10A Problems/Problem 6"

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(deleted first solution, as second solution used same method but elaborated more)
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== Solution ==
 
== Solution ==
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First, we need to note that the digits have to be even, but since only one even digit for the units digit (<math>0</math>) we get <math>4\cdot5\cdot5\cdot1=\boxed{(B)100}</math>.
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The ones digit, for all numbers divisible by 5, must be either <math>0</math> or <math>5</math>. However, from the restriction in the problem, it must be even, giving us exactly one choice (<math>0</math>) for this digit. For the middle two digits, we may choose any even integer from <math>[0, 8]</math>, meaning that we have <math>5</math> total options. For the first digit, we follow similar intuition but realize that it cannot be <math>0</math>, hence giving us 4 possibilities. Therefore, using the multiplication rule, we get <math>4\times 5 \times 5 \times 1 = \boxed{\textbf{(B) } 100}</math>. ~ciceronii
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-middletonkids
 
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== Solution 2 ==
 
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The ones digit, for all numbers divisible by 5, must be either <math>0</math> or <math>5</math>. However, from the restriction in the problem, it must be even, giving us exactly one choice (0) for this digit. For the middle two digits, we may choose any even integer from <math>[0, 8]</math>, meaning that we have <math>5</math> total options. For the first digit, we follow similar intuition but realize that it cannot be <math>0</math>, hence giving us 4 possibilities. Therefore, using the multiplication rule, we get <math>4\times 5 \times 5 \times 1 = \boxed{\textbf{(B) } 100}</math>. ~ciceronii
 
  
 
==Video Solution==
 
==Video Solution==

Revision as of 23:53, 31 January 2020

How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$

$\textbf{(A) } 80 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 125 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 500$

Solution

The ones digit, for all numbers divisible by 5, must be either $0$ or $5$. However, from the restriction in the problem, it must be even, giving us exactly one choice ($0$) for this digit. For the middle two digits, we may choose any even integer from $[0, 8]$, meaning that we have $5$ total options. For the first digit, we follow similar intuition but realize that it cannot be $0$, hence giving us 4 possibilities. Therefore, using the multiplication rule, we get $4\times 5 \times 5 \times 1 = \boxed{\textbf{(B) } 100}$. ~ciceronii

Video Solution

https://youtu.be/JEjib74EmiY

~IceMatrix

See Also

2020 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
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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