Question
If \( a \) and \( b \) are whole numbers and \( a(a-b)=31 \), then \( a+b= \)
Ask by Erickson Nguyen. in the United Arab Emirates
Nov 09,2024
Real Tutor Solution
Tutor-Verified Answer
Answer
The value of \( a + b \) is 61.
Solution
To solve the equation \( a(a-b) = 31 \), we start by rewriting it as:
\[
a^2 - ab = 31
\]
We need to find pairs of whole numbers \( (a, b) \) that satisfy this equation. Since \( 31 \) is a prime number, the possible values for \( a \) must be factors of \( 31 \). The factors of \( 31 \) are \( 1 \) and \( 31 \).
### Case 1: \( a = 1 \)
Substituting \( a = 1 \) into the equation:
\[
1(1 - b) = 31
\]
This simplifies to:
\[
1 - b = 31 \implies -b = 30 \implies b = -30
\]
Since \( b \) is not a whole number, we discard this case.
### Case 2: \( a = 31 \)
Substituting \( a = 31 \) into the equation:
\[
31(31 - b) = 31
\]
This simplifies to:
\[
31 - b = 1 \implies -b = -30 \implies b = 30
\]
Now we have a valid pair \( (a, b) = (31, 30) \).
### Finding \( a + b \)
Now we calculate \( a + b \):
\[
a + b = 31 + 30 = 61
\]
### Conclusion
Thus, the value of \( a + b \) is:
\[
\boxed{61}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To solve the equation \( a(a-b) = 31 \), let's start by recognizing that since \( 31 \) is a prime number, the possible pairs \( (a, a-b) \) that multiply to give \( 31 \) can be either \( (1, 31) \) or \( (31, 1) \) in the domain of whole numbers. 1. **Case 1: \( a = 1 \)** - \( 1(1-b) = 31 \) leads to \( 1-b = 31 \), which gives us \( b = -30 \). This is not a whole number. 2. **Case 2: \( a = 31 \)** - \( 31(31-b) = 31 \) simplifies to \( 31-b = 1 \), resulting in \( b = 30 \). Now, we have possible whole numbers \( a = 31 \) and \( b = 30 \). Calculating \( a + b \): \[ a + b = 31 + 30 = 61 \] Thus, the value of \( a + b \) is \( \boxed{61} \).
