532 More digital trivia
7.Determine the largest and smallest perfect squares which can be
written with the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, used once each
in both cases.8.Find two perfect squares, of five digits each, which together contain
all the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. How many solutions are there
?9.Letk≥ 3. Determine all numbers of the form 1 k+2k+···+nk
whose decimal expansions consist of the ten digits 0, 1, 2, 3, 4, 5,
6, 7, 8, and 9 without repetition.10.Determine all binomial coefficients whose decimal expansions con-
sist of the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 without repetition.
511.Find a number whose cube and fourth power together contain the
ten digits, once each. (See E116 and Problem following E602).^612.Find a multiple of 7 whose square has eight digits of the form
ababbbcc.^713.What is the largest prime whose square contains no duplicate digits
?^814.Find a four-digit square which remains a square when two zeros are
intercalated between the thousands digit and the hundreds digit.^915.Find a number and its fourth power, which together have nine dig-
its, all different.^1016.Find a perfect square whose digits form one of the permutations of
five consecutive digits. (See also E538, 578).^1117.Find a perfect square of 7 digits with all digits even and positive.
12(^5595) 4 ́ and = 5169738420 (^253)
5
́
. = 8301429675
6. = 104976^418 and = 5832^318 18 is the only possible number:In fact, 2, 18 and 69 are the only num-
bers with two distinct powers which together contain the ten digits, once each. 7 = 57577744 2 7588
, even without restriction to multiples of 7.
8. = 457831609^221397
9.^2 700569 = 837, = 7569^287 and^2 200704 = 448, = 2704^252
10. = 1048576^4 n , and=32n
11. = 32041^2179 and = 23104^2152
12. = 8868484^22978 , = 8282884^22878 , = 2862864^21692