Sure, let's break down the solution step-by-step!
The problem states that we have two numbers whose highest common factor (HCF) is 13 and they are in the ratio 4:5. We need to determine the sum of all digits of these two numbers.
Let's denote the first number by [tex]\( a \)[/tex] and the second number by [tex]\( b \)[/tex].
### Steps to find the two numbers:
1. Identify the HCF and Ratios:
- The highest common factor (HCF) given is 13.
- The ratios given are 4 and 5.
2. Express the numbers using the HCF and ratios:
- Since the numbers are in the ratio [tex]\(4:5\)[/tex], we can write them as:
[tex]\[
a = 4 \times 13 = 52
\][/tex]
[tex]\[
b = 5 \times 13 = 65
\][/tex]
3. Sum the digits of the two numbers:
- Sum of the digits of the first number [tex]\(52\)[/tex]:
[tex]\[
5 + 2 = 7
\][/tex]
- Sum of the digits of the second number [tex]\(65\)[/tex]:
[tex]\[
6 + 5 = 11
\][/tex]
- Sum of all digits together:
[tex]\[
7 + 11 = 18
\][/tex]
### Verification for the conditions:
(i) Divisible by 5:
- The sum of all digits is 18, which is not divisible by 5.
(ii) A multiple of 13:
- The sum of all digits is 18, which is not a multiple of 13.
(iii) An odd number:
- The sum of all digits is 18, which is an even number, not odd.
(iv) [tex]\(x^2 + 5x - 4 = 0\)[/tex]:
- To verify this condition, we need to solve the quadratic equation:
[tex]\[
x^2 + 5x - 4 = 0
\][/tex]
Let's use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex] where [tex]\(a = 1\)[/tex], [tex]\(b = 5\)[/tex], and [tex]\(c = -4\)[/tex]:
[tex]\[
x = \frac{-5 \pm \sqrt{25 + 16}}{2} = \frac{-5 \pm \sqrt{41}}{2}
\][/tex]
This results in two irrational numbers and neither relate to the sum 18 directly.
Based on the conditions provided, none of the statements (i), (ii), or (iii) hold true, and statement (iv) is unrelated to the sum of digits 18 directly.
Therefore, the correct step-by-step solution has been provided and aligned with the given problem statement.
