Find all the four digit numbers abba with the property aa.10b=abba. If a is not equal to b.

To solve the problem of finding all four-digit numbers of the form abba that satisfy the equation aa.10b = abba, we need to break down the components of the number and the equation step by step.

Understanding the Structure of the Number

The number abba can be expressed in terms of its digits. Here, 'a' and 'b' are digits, where 'a' is the first and last digit, and 'b' is the second and third digit. Therefore, we can represent abba mathematically as:

• abba = 1000a + 100b + 10b + a = 1001a + 110b

Setting Up the Equation

According to the problem, we also have the equation:

• aa.10b = abba

Here, aa can be expressed as:

• aa = 10a + a = 11a

Substituting this into the equation gives us:

• 11a * 10b = 1001a + 110b

Rearranging the Equation

Now, let's simplify the equation:

• 110ab = 1001a + 110b

Next, we can rearrange it to isolate terms involving 'a' and 'b':

• 110ab - 110b = 1001a
• 110b(a - 1) = 1001a

Solving for 'b'

Now, we can solve for 'b':

• b = (1001a) / (110(a - 1))

At this point, we need to ensure that 'b' is a digit (0 through 9) and that 'a' is not equal to 'b'. We also need to ensure that the denominator is not zero, which means a cannot be 1.

Finding Valid Values for 'a' and 'b'

Let's evaluate possible values for 'a' from 1 to 9 (since 'a' cannot be 0 in a four-digit number) and check if 'b' remains a digit:

• If a = 2: b = (1001 * 2) / (110 * 1) = 20.2 (not a digit)
• If a = 3: b = (1001 * 3) / (110 * 2) = 13.65 (not a digit)
• If a = 4: b = (1001 * 4) / (110 * 3) = 10.09 (not a digit)
• If a = 5: b = (1001 * 5) / (110 * 4) = 11.39 (not a digit)
• If a = 6: b = (1001 * 6) / (110 * 5) = 10.91 (not a digit)
• If a = 7: b = (1001 * 7) / (110 * 6) = 9.57 (not a digit)
• If a = 8: b = (1001 * 8) / (110 * 7) = 8.57 (not a digit)
• If a = 9: b = (1001 * 9) / (110 * 8) = 9.14 (not a digit)

After evaluating all possible values for 'a', we find that none yield a valid digit for 'b' while satisfying the condition that 'a' is not equal to 'b'.

Final Thoughts

In conclusion, there are no four-digit numbers of the form abba that satisfy the equation aa.10b = abba under the condition that 'a' is not equal to 'b'. This exercise illustrates the importance of careful algebraic manipulation and the constraints imposed by digit values in number theory.