### You are given the following equation DOG + DOG + DOG + DOG + DOG + DOG = BRO, where each letter stands for a digit.What is the value of DOG?

105

Step by Step Explanation:
1. We know the following from the question
- Both DOG and BRO are 3 digit numbers.
- The middle digit of the first one is the last digit of the 2nd one. The other digits are different from each other.
- Also, 6 × DOG = BRO
2. Now we need to apply logic to see if we can solve it. Let's see what we can figure out.
3. Hmm...'D' cannot be more than 1, because if it was then 6 × DOG would be a 4 digit number i.e. even the smallest value of DOG in that case would be 200, and 200 × 6 = 1200, which is a 4 digit number.
So, D = 1
4. For the same reason O cannot be more than 6. (As, 170 × 6 = 1020 is a 4 digit number)
5. Now think about G. Can it be even?
No - it cannot be. Why?

6 multiplied by any even number results in the same last digit (6 × 2 = 12, 6 × 4 = 24, 6 × 6 = 36 and so on). But we know that the last digits of DOG and BRO are different.

So G is odd. It can't be 1 though, because D=1. (The value of S and X cannot be same.)
So G either is 3,5,7 or 9.

It cannot be 3, because 3 × 6 = 18. This means O is 8, and we already know O is not greater than 6.

Now it's just a matter of elimination.
6. The possibilities are
G = 5 and O = 0
G = 7 and O = 2
G = 9 and O = 4
7. Now you have 3 possibilities for DOG (105, 127 and 149). It is easy enough to multiply them by 6 and see which one results in the right possibility.
8. 105 × 6 = 630
127 × 6 = 762
149 × 6 = 894
9. 127 × 6 = 762 is not correct, since last digit of DOG should not be same as first digit of BRO.
10. 149 × 6 = 894 is also not correct, since last digit of DOG should not be same as middle digit of BRO.
11. Therefore, correct value of DOG is 105. 