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Find all values of a for which the congruence has integer solutions.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.
The general strategy for this problem will be to evaluate the value of at all integers between 0 and 30 inclusive.
Notice that so evaluating at powers of two should be fairly straight forward.
For other values of x, try to swap between x and its negation to help with the computations. You don't need to compute a lot of values by hand before you can quickly do all 31 values.
Compute it for some small values of primes then use this information along with hint 2 to immediately calculate this for all such values.
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
Clearly and so throughout we suppose that both x is greater than 1 but less than 30. This leaves 28 such values left to be computed. We begin by computing at powers of two. Since
we see that a equals 2 is an admissable value. Further, we immediately get that
and so for b ranging from 1 to 4 we see that a can take on the values 2,4,8 or 16. This covers the x values .
Next for x equals , notice that
which gives us an already chosen a value. Using these facts, we can clearly see that
Further at x equals ,
and this value was also an already chosen a value. I claim that these values are enough to show that all admissable values of a have already been found. To see this, notice for example that
and the right hand side was already shown to be a power of 2. Similarly
is also a power of 2. Crunching through the remaining cases, we have
and in all these cases, we have already shown the right hand side to be a power of 2. Thus the admissable values for a are given by .