Kocour coin values

In Kocourkov, they use coins with only two values expressed in Kocourkov crowns by positive integers. With a sufficient number of such coins, it is possible to pay any integer amount greater than 53 cats’
crowns accurately and without return. However, we cannot pay 53 cat crowns without change.

Find out what values could have been on the cat’s coins. Identify at least two solutions.

Final Answer:

a₁ = 3
b₁ = 28
a₂ = 4
b₂ = 19
a₃ = 7
b₃ = 10
a₄ = 2
b₄ = 55

Step-by-step explanation:

53 \neq = a_{x} + b_{y} 53 < a_{x} + b_{y} a_{1} = 3

b_{1} = 28 t_{1} = 18 \cdot a_{1} + 0 \cdot b_{1} = 18 \cdot 3 + 0 \cdot 28 = 54 t_{2} = 9 \cdot a_{1} + 1 \cdot b_{1} = 9 \cdot 3 + 1 \cdot 28 = 55 t_{3} = 0 \cdot a_{1} + 2 \cdot b_{1} = 0 \cdot 3 + 2 \cdot 28 = 56 \dots b_{1} = 28

a_{2} = 4

b_{2} = 19 t_{4} = 4 \cdot a_{2} + 2 \cdot b_{2} = 4 \cdot 4 + 2 \cdot 19 = 54 t_{5} = 9 \cdot a_{2} + 1 \cdot b_{2} = 9 \cdot 4 + 1 \cdot 19 = 55 t_{6} = 14 \cdot a_{2} + 0 \cdot b_{2} = 14 \cdot 4 + 0 \cdot 19 = 56 t_{7} = 0 \cdot a_{2} + 3 \cdot b_{2} = 0 \cdot 4 + 3 \cdot 19 = 57 b_{2} = \dots = 19

a_{3} = 7

b_{3} = 10

a_{4} = 2

b_{4} = 55 t_{8} = 27 \cdot a_{4} + 0 \cdot b_{4} = 27 \cdot 2 + 0 \cdot 55 = 54 t_{9} = 0 \cdot a_{4} + 1 \cdot b_{4} = 0 \cdot 2 + 1 \cdot 55 = 55 t_{10} = 28 \cdot a_{4} + 0 \cdot b_{4} = 28 \cdot 2 + 0 \cdot 55 = 56 b_{4} = \dots = 55

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