Answers 253carried forward and added to the 4.) But, we cannot do better than take the
lowest multiple (82) of the lowest common multiple of the divisors
(12,252,240), which gives ten figures (this is 1,004,683,680), and keep on
adding that lowest common multiple until all digits are different.
The 199th multiple will give us the first answer, 309 the second, 388 the
third, and 398 the fourth. The work can be considerably shortened by leaping
over groups where figures will obviously be repeated, and all the answers may
be obtained in about twenty minutes by the use of a calculating machine.
- THREES AND SEVENS
The smallest number possible is 3,333,377,733, which is divisible by 3 and
by 7, and the sum of its digits (42) also divisible by 3 and by 7. There must
be at fewest three 7's and seven 3's, and the 7's must be placed as far to the
right as possible.
- ROOT EXTRACTION
The only other numbers are 5832, 17,576, and 19,683, the cube roots of
which may be correctly obtained by merely adding the digits, which come to
18,26, and 27 respectively.- QUEER DIVISION
The smallest number that fulfills the conditions is 35,641,667,749. Other
numbers that will serve may be obtained by adding 46,895,573,610 or any
multiple of it.- THREE DIFFERENT DIGITS
The numbers are 162,243,324,392,405,512,605,648,810, and 972.
These, we think, are all the cases that exist.- DIGITS AND CUBES
There are three solutions. They are 56,169 (the square of 237), where
56 + 69 = 125 (thecubeof5); 63,001 (the square of251), where 63 + 01 =