Answer :
* The equation $5+7=3$ is not valid in standard arithmetic.
* We attempt to find a base $b$ where the equation holds, considering $5_b + 7_b = 3_b$.
* The base $b$ must be greater than 7 because the digits 5 and 7 are used.
* The problem is likely a riddle or trick question, as there's no valid base where $5+7=3$.
### Explanation
1. Understanding the Problem
The equation $5+7=3$ is not correct under standard arithmetic rules. We need to find a base $b$ where this equation holds true.
2. Setting up the Equation
Let's assume that the numbers are in base $b$. Then, the equation can be written as $5_b + 7_b = 3_b$. This implies that $5 + 7 = 3 + kb$, where $k$ is an integer representing the carry-over.
3. Finding Possible Bases
Simplifying the equation, we get $12 = 3 + kb$, which means $9 = kb$. Since $k$ and $b$ are integers, $b$ must be a divisor of 9. The divisors of 9 are 1, 3, and 9.
4. Checking the Base
Since the digits 5 and 7 appear in the equation, the base $b$ must be greater than 7. Therefore, $b=9$. Let's check if the equation holds true in base 9: $5_9 + 7_9 = 12_{10} = 13_9$. This is not equal to $3_9$.
5. Reinterpreting the Equation
However, if we interpret the equation as $5 + 7 = 1 Imes b + 3$, where $b$ is the base, then $12 = b + 3$, which gives $b = 9$. In base 9, $5_9 + 7_9 = 5 + 7 = 12_{10}$. Converting $12_{10}$ to base 9, we get $13_9$. So, $5_9 + 7_9 = 13_9$, not $3_9$.
6. Conclusion
The problem is likely a riddle or a trick question, not a mathematical equation to be solved for a base. There is no mathematical solution in standard number systems.
### Examples
This problem demonstrates that mathematical equations are not always straightforward and can sometimes be puzzles or riddles. It highlights the importance of understanding the context and assumptions behind a problem. Such thinking is useful in cryptography, where codes and ciphers often rely on non-standard interpretations of mathematical operations.
* We attempt to find a base $b$ where the equation holds, considering $5_b + 7_b = 3_b$.
* The base $b$ must be greater than 7 because the digits 5 and 7 are used.
* The problem is likely a riddle or trick question, as there's no valid base where $5+7=3$.
### Explanation
1. Understanding the Problem
The equation $5+7=3$ is not correct under standard arithmetic rules. We need to find a base $b$ where this equation holds true.
2. Setting up the Equation
Let's assume that the numbers are in base $b$. Then, the equation can be written as $5_b + 7_b = 3_b$. This implies that $5 + 7 = 3 + kb$, where $k$ is an integer representing the carry-over.
3. Finding Possible Bases
Simplifying the equation, we get $12 = 3 + kb$, which means $9 = kb$. Since $k$ and $b$ are integers, $b$ must be a divisor of 9. The divisors of 9 are 1, 3, and 9.
4. Checking the Base
Since the digits 5 and 7 appear in the equation, the base $b$ must be greater than 7. Therefore, $b=9$. Let's check if the equation holds true in base 9: $5_9 + 7_9 = 12_{10} = 13_9$. This is not equal to $3_9$.
5. Reinterpreting the Equation
However, if we interpret the equation as $5 + 7 = 1 Imes b + 3$, where $b$ is the base, then $12 = b + 3$, which gives $b = 9$. In base 9, $5_9 + 7_9 = 5 + 7 = 12_{10}$. Converting $12_{10}$ to base 9, we get $13_9$. So, $5_9 + 7_9 = 13_9$, not $3_9$.
6. Conclusion
The problem is likely a riddle or a trick question, not a mathematical equation to be solved for a base. There is no mathematical solution in standard number systems.
### Examples
This problem demonstrates that mathematical equations are not always straightforward and can sometimes be puzzles or riddles. It highlights the importance of understanding the context and assumptions behind a problem. Such thinking is useful in cryptography, where codes and ciphers often rely on non-standard interpretations of mathematical operations.
