# The product of a two-digit natural number and a number consisting of the same digits in the reverse order is 2430. Calculate that two-digit number?

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I know, that the answer is 54 and 45, but how should I calculate it step by step?

I know, that the answer is 54 and 45, but how should I calculate it step by step?

##### 2 Answers

See explanation

#### Explanation:

Well, I didn't know how to solve it absolutely from just what you have in your problem statement. But I made just 1 guess, and the answer flowed from that.

Assume your digits differ by 1. (If this doesn't work, we'll assume they differ by 2, and continue with a similar procedure to what I've done here.)

So, let A be one of the digits. The second will be A + 1.

Then your first number is (A * 10) + (A + 1), and your second is ((A + 1)* 10) + A.

Their product is then:

(10A + A + 1) * (10A + 10 + A) = 2430

...multiply every term in the first set of parens with every term in the second set. Collect terms, and you get:

...which is a quadratic equation with roots 4 and -5.

The digits we want must be "natural" numbers, i.e., positive integers. So disregard the -5, and pick A = 4. The next digit must be 5. (Remember, we made the single assumption that they differed by 1.)

Which would give 45 and 54.

(I went through a similar procedure with the assumption that the 2 digits differed by 2. Your resulting quadratic equation is

The numbers are

#### Explanation:

The last digit of

This indicates that one of the numbers must end in a

Let the missing digit be

The number

The number

The product is

Using

Find factors of

If

Check