MathLabs

Problem 1

Find all integers a,b,c satisfying 1<a<b<c1<a<b<c such that (a−1)(b−1)(c−1)(a-1)(b-1)(c-1) is a divisor of abc−1abc-1.
Step 2 of 6: The size bound forces a≤4
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a≥5 ⟹ 2(a−1)(b−1)(c−1)>abc>abc−1 ⟹ a≤4a\ge5\ \Longrightarrow\ 2(a-1)(b-1)(c-1)>abc>abc-1\ \Longrightarrow\ a\le4
Detailed analysis

If a≥5, then b≥a+1 and c≥a+2. The ratio of the product (a−1)(b−1)(c−1) to abc is at least (4/5)(5/6)(6/7)=4/7>1/2. Hence twice that product exceeds abc, so the positive integer q must be 1. But the q=1 case is impossible by the next step. Therefore a≤4; equivalently, directly the quotient cannot be at least 2 in this range.