Let the three consecutive numbers be assume that x, (x + 1), (x + 2) respectively.
Given that x, (x + 1), (x + 2) are divided by 10, 17, 26 respectively, the sum of the quotients is 10.
Then
⇒ \(\frac x{10}\,+\,\frac{x\,+\,1}{17}\,+\,\frac{x\,+2}{26}\) = 10
⇒ \(\frac{x\,\times\,221\,+\,130(x\,+\,1)\,+\,85(x\,+\,2)}{2210}\) = 10
⇒ 221x + 130x + 85x + 130 + 170 = 22,100
⇒ 436x + 300 = 22,100
⇒ 436x = 22,100 – 300
⇒ 436x = 21,800
⇒ \(\frac{21800}{436}\)
∴ x = 50
∴ The required three consecutive numbers are x = 50
x + 1 =50 + 1 = 51
x + 2 = 50 + 2 = 52
