Henry Ernest Dudeney/Puzzles and Curious Problems/110 - An Absolute Skeleton/Solution 2

by : $110$

 * An Absolute Skeleton

Solution
32819764 310)10174126840     930       ---       874       620       2541       2480       -         612         310         3026         2790         -          2368          2170          -           1984           1860           -            1240            1240

Proof
From the initial deductions, we have determined that:
 * $(1): \quad$ It is possible that there is a solution in which the divisor is $310$
 * $(2): \quad$ If this is the case, then the quotient is one of:
 * $32418697$
 * $32419786$
 * $32816497$
 * $32819764$

Thus the dividend in each case will be:
 * $10049796070$
 * $10050133660$
 * $10173114070$
 * $10174126840$

It remains to investigate each one.

Hence we set up a long division and check the partial dividends in each case.

32418697 310)10049796070     930       ---       749       620       1297       1240       -         579         310         2696         **** and we need go no further.

32419786 310)10050133660     930       ---       750       620       1301       1240       -         613         310         3033         ****         - and we need go no further.

32816497 310)10173114070     930       ---       873       620       2531       2480       -         511         ***

and we need go no further.

Finally:

32819764 310)10174126840     930       ---       874       620       2541       2480       -         612         310         3026         2790         -          2368          2170          -           1984           1860           -            1240            1240 and we have found a solution.

Hence the only solution with $D = 310$ is:


 * $\dfrac {10174126840} {310} = 32819764$