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This is basically the Frobenius/coin problem.
Use the Frobenius coin problem result for two coprime integers a and b: the largest non-representable integer is ab - a - b, so the smallest N such that all integers ≥ N are representable is ab - a - b + 1. For a=5 and b=21, compute 5*21 - 5 - 21 + 1 = 80. Explain the reasoning and verify with a small example or modular argument.
Pro tip: Mention that this is a classic number theory result (Frobenius number) and that for two coprime numbers the formula is exact; for more than two numbers no simple closed form exists, which shows depth. Also, briefly verify that 79 is not representable (e.g., 79 = 21*3 + 16, and 16 cannot be made with 5s and 21s) to confirm N=80.
Recognize that this is the Frobenius coin problem for two coprime integers. State that since gcd(5,21)=1, all sufficiently large integers are representable.
For two coprime positive integers a and b, the largest non-representable integer is ab - a - b. Therefore the smallest N such that all integers ≥ N are representable is ab - a - b + 1.
Plug in a=5, b=21: ab - a - b + 1 = 105 - 5 - 21 + 1 = 80. So N=80.
Show that 79 is not representable (e.g., 79 = 21*3 + 16, and 16 cannot be formed by 5s and 21s) and that 80 is representable (80 = 5*16). This confirms the boundary.
Mention that for more than two numbers, the Frobenius number is not known in closed form and is NP-hard to compute, which is relevant in algorithm design and data science contexts.
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