A two digit number is k times the sum of the digit. By interchanging the digits, Find the number is how many times its sum of the digits. Explain method
To find the number of times a two-digit number is equal to the sum of its digits after interchanging the digits, we can follow these steps:
1. Let's assume that the digits of the two-digit number are represented as "a" and "b". So, the original number can be written as 10a + b.
2. According to the given information, the number is k times the sum of its digits. The sum of the digits can be calculated as a + b. Therefore, we have the equation:
10a + b = k(a + b)
3. Now, let's simplify the equation by expanding it:
10a + b = ka + kb
4. Next, let's rearrange the terms to isolate "b" on one side of the equation:
10a - ka = kb - b
(10 - k)a = (k - 1)b
5. Since a and b are integers, (10 - k) and (k - 1) must have a common factor. Let's assume their common factor is "m". So, we have:
(10 - k) = m
(k - 1) = m
6. Now, let's solve these two equations simultaneously to find the values of k, m, a, and b.
From equation (1): (10 - k) = m
From equation (2): (k - 1) = m
Solving these two equations, we get:
k = 9 - m
7. Since k represents the tens digit of the number, it should be between 1 and 9 (inclusive). Therefore, there are nine possible values for m, which are 1, 2, 3, 4, 5, 6, 7, 8, and 9.
8. For each value of m, we can calculate the corresponding value of k using k = 9 - m.
9. Once we have the value of k, we can find the original number (10a + b) by substituting the value of k into equation (1) or (2).
10. After finding the original number, we can interchange the digits and calculate the sum of the digits.
11. Finally, we can determine how many times the original number is equal to the sum of its digits by dividing the original number by the sum of its digits.
By following these steps, you can find the number of times a two-digit number is equal to the sum of its digits after interchanging the digits for any given value of "k".