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A number is divisible by 8 if the number formed by its last 3 digits is divisible by 8. For example, the number 47,389,248 is divisible by 8 because 248 is divisible by 8. However, 47,389,284 is not divisible by 8 because 284 is not divisible by 8.

If 992,466,1A6 is divisible by 8, where A represents one digit, what is the sum of the possible values of A?

 Mar 1, 2021
 #1
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A = 3  and   A= 7

 

3   +  7 ==10

 Mar 1, 2021
 #2
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like you said, "A number is divisible by 8 if the number formed by its last 3 digits is divisible by 8." so, we have to find the lowest possible number there is for 1A6, which would be 136 because 96 is divisibe by 8, and the smallest number that is divisible by both 8 and 10 is 40, so we add 40. then, we can repeat that process once more, and we get 136, and 176. there's no other numbers that have hundreds digit of 1, and units digit as 6 toher than those two, so 3+7=$\boxed{10}$

 Mar 1, 2021
 #3
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The problem even explains how to determine whether or not a given number is divisible by 8. Only the final three digits matter, so determining if 992,466,1A6 is divisible by 8 is the same question as asking if 1A6 is divisible by 8.

 

Let's check all the digits.

Is 116 divisible by 8? No.

Is 126 divisible by 8? No.

Is 136 divisible by 8? Yes!

 

Of course, I could keep going and check the rest of the digits, but I have a small shortcut that saves a little bit of time. Since we are changing the tens digit, the next number that will be divisible by 8 again will be the current number + 40. Why? Well, 40 is divisible by 8, as well, so adding 40 to a number divisible by 8 will still yield a number divisible by 8. 

 

136 is divisible by 8

176 is divisible by 8

216 is divisible by 8, but that is not allowed since A is a digit, and digits range from 0-9.

 

Therefore, when A = 3 and A = 7, 992,466,1A6 is divisible by 8.

 

The sum of all possible A's is 3+7 = 10.

 Mar 1, 2021

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