Question

If you multiply the four-digit number abcd by 4, the order of digits will be reversed. That is, abcd x 4=dcba. The digits a, b, c and d are all different. Find abcd. That means what numbers are abcd equal to?

Answer 1

The number is 2178, multiplied by 4 gives 8712.

Explanation:

Let us say the number is written as abcd
`(1000a+100b+10c+d)` .....(1)

When you multiply it by 4, it gives dcba
`(1000d+100c+10b+a)`

Now,
`4*(abcd)=dcba`

We get;

`4000a+400b+40c+4d =1000d+100c+10b+a`

=>
`3999a+390b=996d+60c`

=>
`1333a+130b= 20c+332d` ......(2)

Now, we can relate few properties:

The first digit is a quarter of the last one as d=4a .

or
`a=d/4`

or
`a=0.25d`

The second digit is one less than the first.

`b=(a-1)`

Now, from equation 2, we get

`1333a+130b= 20c+332d`

Replacing b and d, we get

`1333a+130a-130=20c+1328a`

=>
`135a=20c+130`

=>
`27a=4c+26`

or,
`a =(4/27)c+(26/27)`

There is only one single digit integer solution.

c = 7; a = 2

Now,

`b =2-1`=1 and d =4(2) =8

Hence, 2178 is the number.

And we can see that
`4*2178= 8712`

Answer 2

The given is that A B C D x 4 = D C B A

Possible values of A are 1 or 2. It can't be 1 since the final result A is in unit place and 1 is not possible when we multiply any number by 4

2 B C D x 4 = D C B 2
It is clear above, that D = 8
2 B C 8 x 4 = 8 C B 2
Possible values of B should be 1 or 2
We try B=1

2 1 C 8 x 4 = 8 C 1 2

To find for C, you can use the equation:

(2000+100+10C+8) x 4 = 8000+100C+10+240C = 432-12 = 420
Therefore,

C = 7
So, the number is 2178.