<p>What is the largest amount that can not be paid by using notes of denomination ₹5 and ₹7.</p>
Step-by-Step Solution
Key Concept: For coprime integers a and b, the largest number that cannot be expressed as a non-negative integer linear combination is ab - a - b. Here, gcd(5,7)=1, so the answer is 5×7 - 5 - 7 = 23.
<p><strong>Step 1:</strong> Recognize this as the Frobenius coin problem. We need to find the largest amount that <u>cannot</u> be expressed as 5m + 7n where m,n ≥ 0 are non-negative integers.</p><p><strong>Step 2:</strong> Since gcd(5,7) = 1 (coprime), the Frobenius number formula applies: F(a,b) = ab - a - b</p><p><strong>Step 3:</strong> Calculate F(5,7) = (5)(7) - 5 - 7 = 35 - 12 = 23</p><p><strong>Step 4:</strong> Verify: 23 cannot be made (5m + 7n = 23 has no non-negative integer solutions), but 24 = 5(1) + 7(1) + 7(1) + 5(0)... Actually: 24 = 5(1) + 19? Check: 24 = 17 + 7 ✗. Try: 24 = 5(5) - 1 ✗. Better: Any amount ≥ 24 can be made (this requires checking or using the theorem that all integers ≥ F(a,b)+1 are achievable).</p><p>∴ Answer: <strong>23</strong></p>
Correct Answer: 23