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 3 of 5: When n is divisible by 4, the gap is 2
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n=4m ⟹ 2m−1,2m+1 reduced ⟹d∣2n=4m\ \Longrightarrow\ 2m-1,2m+1\text{ reduced}\ \Longrightarrow d\mid2
Detailed analysis

For n=4m, both 2m−1 and 2m+1 are odd and have gcd 1 with n. Their difference is 2, so d divides 2. Every listed number is odd, hence d cannot be 1; therefore d=2 and the list is exactly all odd numbers below n.