ABCDE × 4 = EDCBA

Number Theory: Reverse digits by multiplying by 4

Find two five-digit numbers such that
one is 4 times the other, and the digits
of the first are the reverse of the digits of the second.
In other words, ABCDE × 4 = EDCBA?

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Since EDCBA is a 5-digit number, we know ABCDE is < 1/4 × 100000.
ABCDE < 25000, so A is 1 or 2.

But EBCDA is a multiple of 4, so it is even, so A = 2.
4 * ....E = ....2, so E is either 3 or 8.
But, 4 × 2nnnn cannot be 3nnnn so E is 8.

We have 2BCD8 * 4 = 8DCB2.
and therefore 4 × BCD + 3 = DCB.
Let's look at it case by case, for values of D:
D = 1, B = 7
D = 2, B = 1
D = 3, B = 5
D = 4, B = 9
D = 5, B = 3
D = 6, B = 7
D = 7, B = 1
D = 8, B = 5
D = 9, B = 9
D = 0, B = 3

But B cannot be greater than 2, or there would be a carry, and B cannot
be equal to 2, since A = 2 (and we assume each letter represents a
different number).
So, B = 1 and D = 2 or 7.
But again, A is already 2, so D = 7.

So we have 21C78 * 4 = 87C12
We now have 4 * C + 3 = C (mod 10)
3*C = 7 mod 10
and therefore C = 9

So the answer is 21978 * 4 = 87912