Henry Ernest Dudeney/Puzzles and Curious Problems/110 - An Absolute Skeleton/Solution 1

by : $110$

 * An Absolute Skeleton

Solution
32418675 312)10114626600     936       ---       754       624       1306       1248       -         582         312         2706         2496         -          2106          1872          -           2340           2184           -            1560            1560

Proof
From the initial deductions, we have determined that:
 * $(1): \quad$ It is possible that there is a solution in which the divisor is $312$
 * $(2): \quad$ If this is the case, then the quotient is one of:
 * $32417586$
 * $32418675$
 * $32816475$
 * $32817564$

Thus the dividend in each case will be:
 * $10114286832$
 * $10114626600$
 * $10238740200$
 * $10239079968$

It remains to investigate each one.

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

32417586 312)10114286832     936       ---       754       624       1302       1248       -         548         312         2366         **** and we need go no further.

32418675 312)10114626600     936       ---       754       624       1306       1248       -         582         312         2706         2496         -          2106          1872          -           2340           2184           -            1560            1560 and we have found a solution.

Let us try the remaining solutions:

32816475 312)10238740200     936       ---       878       *** and we need not continue with this one.

Finally:

32817564 312)10239079968     936       ---       879       624       2550       **** and again it is seen that this also does not fit the criteria.

Hence the only solution with $D = 312$ is:


 * $\dfrac {10114626600} {312} = 32418675$