Question

If m and n are two positive integers such that \( 7m + 11n = 200 \), then the minimum possible value of m + n is ________

Hint:

For equations of the form \(ax + by = c\), to minimize \(x+y\) where \(a<b\), you should maximize the variable with the larger coefficient (in this case, n). This reduces the value of the other variable more quickly, generally leading to a smaller sum.

Answer 1

Correct Answer: 20

Step 1: Understanding the Concept:
This is a problem involving a linear Diophantine equation, where we need to find integer solutions. Since we are looking for the minimum value of a sum, we should test the possible integer solutions.
Step 2: Key Formula or Approach:
1. Isolate one variable in the equation \( 7m + 11n = 200 \). 2. Use the constraints that m and n must be positive integers to find the range of possible values for one variable. 3. Test the possible values to find integer pairs (m, n) that satisfy the equation. 4. Calculate the sum m+n for each pair and find the minimum.
Step 3: Detailed Explanation:
We are given the equation \( 7m + 11n = 200 \), where m and n are positive integers. Let's express m in terms of n: \( m = \frac{200 - 11n}{7} \). Since m must be a positive integer, \( 200 - 11n>0 \implies 11n<200 \implies n \le 18 \). Also, \(200 - 11n\) must be divisible by 7. We can test values of n from 1 to 18. Alternatively, using modular arithmetic: \( 200 - 11n \equiv 0 \pmod{7} \). \( 200 \equiv 4 \pmod{7} \) and \( 11 \equiv 4 \pmod{7} \). So, \( 4 - 4n \equiv 0 \pmod{7} \implies 4n \equiv 4 \pmod{7} \). Since gcd(4, 7) = 1, we can divide by 4, which gives \( n \equiv 1 \pmod{7} \). This means n must be of the form \(7k+1\). Possible values for n are 1, 8, 15. Let's test these values:

If \( n = 1 \): \( 7m = 200 - 11(1) = 189 \implies m = 27 \). Sum \( m+n = 27+1 = 28 \).
If \( n = 8 \): \( 7m = 200 - 11(8) = 112 \implies m = 16 \). Sum \( m+n = 16+8 = 24 \).
If \( n = 15 \): \( 7m = 200 - 11(15) = 35 \implies m = 5 \). Sum \( m+n = 5+15 = 20 \).
The possible sums are 28, 24, and 20. The minimum value is 20.
Step 4: Final Answer:
The minimum possible value of m + n is 20.