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I'm just sharing my approach here because I think it's a bit more algorithmic than others. So we have 3 questions to ask a potential solution (divisible by 5, 8, 10?) - if we make zero assumptions about what is possible then the set of answers can be 8 possible sets. The truth table below sets out the possible answers, and then from there we can infer three ...


Summarizing the hints: What are the options? Is it a multiple of 10? So


As we know, As So we know that So the result is that the last digit for $3^{2019}$ is:


Because and we have so the answer is



He has Let M be the number of memory cards: If M is between 1-19 then it could be any number, however If M is between 20-29 then it is a not a multiple of 8. Which means that is could be 20, 21, 22, 23, 25, 26, 27, 28 or 29. If M is between 30-39, then it's not a multiple of 10. Which means it could be 31-39. Therefore the only number it can be is ...


I don't think a definitive answer can be reached, I did see Lanny's answer but I don't agree with the logic above to find (9, 15) as the pair and this is my answer. Lanny's answer does not include 135, pairs being (9, 15) and (3, 45), or 140 with the pairs being (10, 14) and (2, 70). I don't see the reasoning to eliminate these. I think the right logic is to ...


The numbers are: Step 1: Step 2 Step 3 Step 4 Step 5


First of all, let's see why your brute-forcing fails. (This is the puzzle part, the rest is plain old math.) No matter which you chose, the number at the bottom right would have to be both odd and even at the same time, so there's no integer solution. However, there are four equations and four unknowns, so we should have at least one solution (unless the ...

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