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Find a four-digit number from the difference with its reverse

Write the digits as place values

If the digits are $a, b, c, d$ , write the number as $1000 a + 100 b + 10 c + d$ and its reverse as $1000 d + 100 c + 10 b + a$ . Then use the divisibility conditions before choosing digits.

Problem

The digits of a four-digit multiple of 5 are reversed to make another four-digit number. Subtracting the reversed number from the original gives 3627. Give exactly one possible original number.

Answer

One possible number is

8705

Its reverse is 5078, and $8705-5078=3627$.

Step-by-step solution

1. Fix the last digit

A multiple of 5 ends in 0 or 5. The reversed number must still have four digits, so the last digit cannot be 0. It is 5. Write the original as $1000 a + 100 b + 10 c + 5$ , where $a$ is a nonzero digit.

2. Use the difference

The reversed number is $5000 + 100 c + 10 b + a$ . Subtraction gives

999 a + 90 (b - c) = 8622.

Reducing this equation modulo 90 gives $9 a \equiv 72 (mod 90)$ . As $a$ is a digit from 1 to 9, $a = 8$ . The equation then becomes $90 (b - c) = 630$ , so $b - c = 7$ .

3. Choose and check one pair

Take $b = 7$ and $c = 0$ . The original number is $8705$ and its reverse is $5078$ . Their difference is $3627$ , as required.

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