MathLabs

Problem 2

Let n>6 be an integer and let a1,a2,…,aka_1,a_2,\ldots,a_k be all positive integers less than n and relatively prime to n, in increasing order. If a2−a1=a3−a2=⋯=ak−ak−1>0a_2-a_1=a_3-a_2=\cdots=a_k-a_{k-1}>0, prove that n is either a prime or a power of 2.
Step 4 of 5: The divisible-by-4 case gives a power of 2
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n=4m and all odd r<n are reduced ⟹ n=2sn=4m\text{ and all odd }r<n\text{ are reduced}\ \Longrightarrow\ n=2^s
Detailed analysis

If n had an odd prime divisor q, then q<n would be an odd number in the list, but q would not be relatively prime to n. Thus n has no odd prime divisor and is a power of 2.