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 4 of 6: The case q=2 gives (3,5,15)

Previous step Next step

q=2\ \Longrightarrow\ a,b,c\text{ odd}\ \Longrightarrow\ a=3,\qquad bc+5=4b+4c

Detailed analysis

For q=2, abc−1 is even, so abc is odd and all three variables are odd. Since a≤4 and a>1, a=3. The equation becomes bc+5=4b+4c. If b≥9, then bc≥9c>4b+4c, impossible; hence b=5 or 7. These give c=15 and c=23/3 respectively, so only (3,5,15) works.