Question

已知正整数a,b,c满足a<b<c,且1/(a-1)+1/(b-1)+1/(c-1)=1,求a,b,c的值

Answer 1

若a > 3, 则a ≥ 4, b ≥ a+1 ≥ 5, c ≥ b+1 ≥ 6.
1/(a-1)+1/(b-1)+1/(c-1) ≤ 1/3+1/4+1/5 < 1, 故a ≤ 3.
若a = 2, 则1/(a-1)+1/(b-1)+1/(c-1) > 1/(a-1) = 1.
故只有a = 3.
代回得1/(b-1)+1/(c-1) = 1/2, 整理得bc-3b-3c+5 = 0.
即(b-3)(c-3) = 4.
由整数c > b > a = 3, b-3与c-3都是4的正约数, 且c-3 > b-3.
只有b-3 = 1, c-3 = 4.
解得b = 4, c = 7.
故唯一满足要求的解为(a,b,c) = (3,4,7).