Number Theory

N == 219,978  

879,912 ==4 * 219,978

4(abcdef)=fedcba

From the LHS  a could be 1  or 2   but it can't be 1 because nothing times 4 has the last digit 1

SO

a is 2    which means f is 8 or 9 but it can't be 9 because nothing times 4 has the last digit 1

SO

a=2   and   f=8

4(2bcde8)=8edcb2

SO

4(bcde8)= edcb2

b has to be  0, 1 or 2

4*8=32

so

4*e+3 = #b    b must be 0,1,or 2

so

4*e+3= #0. #1,   or #2

4*e    = #7, #8,   or  #9

the only one that works is

4*e = #8

4*2 is 8 and 4*7 is 28    so  e must be  2 or 7   but the smallest e can be is 4 

SO

e=7

4(bcd78)= 7dcb2

Looking at that I can see that b must be 0 or 1 becaust anthing else is too big .. but 0 is too small

SO

b=1

4(1cd78)= 7dc12

subtract 40000 off both sides and you get

4(cd78)= 3dc12

now 4*78 = 312

so

4*(10c+d)+3 = 300+10d+c

40c+4d+3=300+ 10d+c

39C=6d+297

So  C=9 and D=9

4(9978)= 39912    that works

4(219978)= 879912   

N is the smaller one so

N=219978

Just as guest found with a computer program :)   Thanks answer Guest.