Find the greatest number which divides 34,60 and 85 leaving remainders of 7,6 and 4 respectively.

Since 7, 6 and 4 are the remainders of 34, 60 and 85 respectively.
So required number should divide (34 - 7) i.e. 27; (60 - 6) i.e. 54 and (85 - 4) i.e. 81 completely.
Thus required number = H.C.F of 27, 54 and 81.
Now we have;
$$\begin{gathered} 27 = 3\times 3\times 3 \\ 54 = 3\times 3\times 3\times 2 \\ 81 = 3\times 3\times 3\times 3 \\ \mathrm{So}; \mathrm{H}.\mathrm{C}.\mathrm{F} \mathrm{of} 27, 54 \mathrm{and} 81 = 3\times 3\times 3 = 27 \end{gathered}$$
Therefore the required number is 27.