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Number Theory Level 4

Let a , b , c a,b,c be non-negative integers. What is the maximum possible value of c c such that 7 a + 12 b = c 7a + 12b = c has no solution?

Bonus: Generalize this.


The answer is 65.

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Sathvik Acharya by Sathvik Acharya , 17, India

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1 solution

Michael Mendrin
Feb 10, 2017

Given positive numbers x , y x, y such that x a + y b = c xa+yb=c has no solution, the largest possible value of c c is x y − x − y xy-x-y , provided that a , b a, b are positive coprimes. In this case, that works out to

7 ⋅ 12 − 7 − 12 = 65 7\cdot12-7-12=65

2 Helpful 0 Interesting 0 Brilliant 0 Confused

the largest possible value of c c is x y − x − y xy-x-y

Why is this true? Shouldn't you specify that x and y are positive integers as well?

Pi Han Goh - 4 years, 3 months ago

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