To find the two-digit number that meets the given conditions, let's go through the problem step-by-step:
1. Understand the Problem:
- We have a two-digit number where the digits multiply to 24.
- When the digits of this number are reversed, the resultant number is greater than the original number by 18.
2. Setup Variables:
- Let the original number be represented as [tex]\(10a + b\)[/tex] where 'a' is the tens digit and 'b' is the units digit.
- Therefore, the reversed number would be [tex]\(10b + a\)[/tex].
3. Translate Conditions into Equations:
- From the given conditions, we can form the following two algebraic equations:
1. The product of the digits is 24, which can be written as:
[tex]\[
a \times b = 24
\][/tex]
2. When the digits are reversed, the new number is 18 more than the original number:
[tex]\[
10b + a = 10a + b + 18
\][/tex]
4. Simplify the Second Equation:
- Rearrange [tex]\(10b + a = 10a + b + 18\)[/tex] to get:
[tex]\[
10b + a - 10a - b = 18
\][/tex]
Simplify to:
[tex]\[
9b - 9a = 18
\][/tex]
Divide by 9:
[tex]\[
b - a = 2
\][/tex]
5. Solve the System of Equations:
- We now have two equations:
1. [tex]\(a \times b = 24\)[/tex]
2. [tex]\(b - a = 2\)[/tex]
- Solving [tex]\(b - a = 2\)[/tex] for [tex]\(b\)[/tex]:
[tex]\[
b = a + 2
\][/tex]
- Substitute [tex]\(b = a + 2\)[/tex] into [tex]\(a \times b = 24\)[/tex]:
[tex]\[
a \times (a + 2) = 24
\][/tex]
Expand and solve the quadratic equation:
[tex]\[
a^2 + 2a - 24 = 0
\][/tex]
- Factoring the quadratic equation:
[tex]\[
(a + 6)(a - 4) = 0
\][/tex]
- Thus, the solutions for [tex]\(a\)[/tex] are:
[tex]\[
a = -6 \quad \text{or} \quad a = 4
\][/tex]
6. Find Corresponding Values for [tex]\(b\)[/tex]:
- If [tex]\(a = 4\)[/tex], then:
[tex]\[
b = 4 + 2 = 6
\][/tex]
7. Form the Original Number:
- Substituting [tex]\(a = 4\)[/tex] and [tex]\(b = 6\)[/tex] into the original number format [tex]\(10a + b\)[/tex]:
[tex]\[
\text{Original Number} = 10 \times 4 + 6 = 46
\][/tex]
8. Check the Solution:
- The product of the digits is:
[tex]\[
4 \times 6 = 24
\][/tex]
- The reversed number is [tex]\(64\)[/tex], and [tex]\(64\)[/tex] is indeed [tex]\(18\)[/tex] more than [tex]\(46\)[/tex]:
[tex]\[
64 - 46 = 18
\][/tex]
Hence, the original two-digit number is [tex]\(46\)[/tex].
