Problem 1

Find all integers a,b,c satisfying

1<a<b<c

such that

(a-1)(b-1)(c-1)

is a divisor of

abc-1

Step 5 of 6: For a=2, only q=3,4,5 need checking

Previous step Next step

a=2,\ q\ge3:\quad (q-2)bc+(q+1)=q(b+c)

Detailed analysis

Substituting a=2 gives (q−2)bc+(q+1)=q(b+c). Since b≥3, for q≥6 the left term (q−2)bc is at least (3q−6)c and hence at least 2qc, which exceeds q(b+c) because c>b. Thus q is 3,4, or 5. For q=3, the equation is bc+4=3b+3c; b≥6 is impossible, and b=3,4,5 leaves only b=4,c=8. For q=4 parity rules out solutions. For q=5, the equation is 3bc+6=5b+5c; b=3 gives c=9/4 and b≥4 is impossible by size.