14 Angry Numbers!
a 1 4 + a 2 4 + a 3 4 + a 4 4 + … + a 1 4 4 = 4 9 9 9 9
Let there be integers a 1 , a 2 , a 3 , … a 1 4 , such that the above equation is satisfied.
How many 1 4 − tuples of a 1 , a 2 , a 3 … , a 1 4 exist such that the above condition is met?
The answer is 0.
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1 solution
Great solution. Can you tell me how to learn to analyse such techniques? (Finding which mod to take) @Bogdan Simeonov
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Well, if you are familiar with the phi function, you should look for numbers whose totient is the power.For instance, the power here is 4, and phi of 16 is equal to four, so it works.
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Thanks for adding that in. Useful :D.
hey can we use it in finding integral solutions like for how many triplets x 1 + x 2 + x 3 =56
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@Rishabh Jain – No, in that case we can't use it.
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@Bogdan Simeonov – so how can we solve it ??
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@Rishabh Jain – We can use Stars and Bars technique!!
how did you come up with 16?
can you please cross check if the euler function of 16 is 4 or 8? @Bogdan Simeonov
Weird to have a question start with "let there be...." when no solutions exist.
The same could be done mod 5, which gives 0 or 1 remainder. Yep there we have to eliminate 3 cases.
Looking mod 16, a 4 gives remainder 1 or 0(That can be verified by looking at all the cases mod 16).But RHS is congruent to 15(mod 16), and the LHS is a sum of 14 0s and 1s which is impossible.