if you multiply a 4 digit number abcd by 4, the digits get reversed. that is abcd x 4 dcba. find abcd?

Question

whoa confusing help!!

Answer 1

2178
x 4
====
8712

Figure this out by elimination. Start with "a". "a" must be 1 or 2 because the final answer is 4 digits long. If "a" > 2 then the final answer would be 5 digits long. So
a = 1 or 2. If a is 1 then 4 * d must end with a 1. This isn't possilbe. So a must be 2.

2bcd
x 4
====
dcb2

so d has to be 3 or 8 (only digits that yield a 2 on the bottom).
d has to be 8 because 4 x 2 will yield either 8 or 9 (if you carried a 1 from the previous multiplication). 9 isn't valid for d so:

d = 8

2bc8
x 4
====
8cb2

now, this means that b x 4 can't be > 10 because we didn't carry anything to the 4 x 2. So b must be 1 or 2.

Now try all the c and b values that yield a 1 or 2 for b. This is (c x 4) + 3 (the carry from 8 x 4).
c=0,1,2,3,4,5,6,7,8,9
b=3,7,1,5,9,3,7,1,5,9

we said that b must be 1 or 2. c must be either 2 or 7. Try them both and you will find the answer.

Answer 2

Let the original number, n, be 1000a + 100b+10c + d.
Then 4(1000a+100b+10c+d) = 1000d+100c+10b+a (*)
This gives us that 4d = a(mod 10) so a is even
Also, we can assume that a isn't 0.
Since 4n is a 4 digit number, a < 3, so a = 2.
Plugging this into (*), we get
7998 = 996d + 60c -390b.
Next, b < 5 because 4n is a 4 digit number.
If b were even, the right side of this equation
would be divisible by 4, but not the left side.
So b is odd and b = 1 or 3.
If b = 1, we get 2097 = 249d + 15c,
and, trying all the possibilities for c and d,
we find c = 7, d = 8.
So a number that fills the bill is 2178
and 2178*4 = 8712.
No solutions exist for b = 3.