How many 10-digit positive integers with distinct digits are multiples of 11111?

Answer 1

-11101

Answer 2

Since digits of 10digits number n sum to 45 , n should be divisible by 9

n should be divisible by 99999

10 digits number will be of type n = 100000*x + y where x is 5digit number and y is 4 or 5 digit number

n = 99999*x + (x+y)

means x+y should be divisible by 99999

and because neither x > 99999 nor y>99999

x+y = 99999

x=abcde

y=pqrst

----------------------

x+y=99999

also since max numbers is 9 and 8 we cannot carry to next digit (19)

so it has to be all 9s without carry

groups

0 9

1 8

2 7

3 6

4 5

means combination is for x only. y is related

digit 1 : {1-9} 9ways

digit 2 : {digit1,0} - {digit1,9-digit1} 8 ways

digit 3 : {digit2} - {digit2.9-digit2} 6 ways

digit 4 : {digit3} - {digit3.9-digit3} 4 ways

digit 5 : {digit4} - {digit4.9-digit4} 2 ways

total ways : 9*8*6*4*2=3456