more games of life

Another puzzle in memoriam of John Conway in The Guardian:

Find the ten digit number, abcdefghij. Each of the digits is different, and

• a is divisible by 1
• ab is divisible by 2
• abc is divisible by 3
• abcd is divisible by 4
• abcde is divisible by 5
• abcdef is divisible by 6
• abcdefg is divisible by 7
• abcdefgh is divisible by 8
• abcdefghi is divisible by 9
• abcdefghij is divisible by 10

Which brute force R coding by checking over random permutations of (1,2,…,9) [since j=0] solves within seconds:

while(0<1)
  if (prod(!(x<-sum(10^{0:8}*sample(1:9)))%/%10^{7:0}%%2:9))break()

into x=3816547290. And slightly less brute force R coding even faster:

while(0<1){
e=sample(c(2,6,8))#even
o=sample(c(1,3,7,9))#odd
if((!(o[1]+e[1]+o[2])%%3)&
(!(10*o[2]+e[2])%%4)&
(!(o[1]+e[1]+o[2]+e[2]+5+4)%%3)&
(!sum(10^{6:0}*c(o[1],e[1],o[2],e[2],5,4,o[3]))%%7)&
(!(10*o[3]+e[3])%%8)&
(!(sum(o)+sum(e))%%9)){
print(sum(10^{9:0}*c(o[1],e[1],o[2],e[2],4,5,o[3],e[3],o[4],0)));break()}}