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 3 of 6: Exclude q=1
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q=1 ⟹ a+b+c=ab+bc+ca(impossible)q=1\ \Longrightarrow\ a+b+c=ab+bc+ca\quad\text{(impossible)}
Detailed analysis

Expanding the equation for q=1 gives a+b+c=ab+bc+ca. Since all variables exceed 1, we have a<ab, b<bc, and c<ca, so the left side is strictly smaller than the right side, a contradiction.