Banknotes

How many different ways can the cashier payout € 310 if he uses only 50 and 20 euro banknotes? Find all solutions.

Result

n =  3

Solution:

310=50a+20b 310=2⋅5⋅31 50=2⋅52 20=22⋅5 GCD(310,50,20)=2⋅5=10  d=GCD(310,50,20)=10  31=5a+2b  a1=1,b1=13 a2=3,b2=8 a3=5,b3=3  n=3310=50a + 20b \ \\ 310=2 \cdot 5 \cdot 31 \ \\ 50=2 \cdot 5^2 \ \\ 20=2^2 \cdot 5 \ \\ \text{GCD}(310, 50, 20)=2 \cdot 5=10 \ \\ \ \\ d=GCD(310,50,20)=10 \ \\ \ \\ 31=5a + 2b \ \\ \ \\ a_{1}=1, b_{1}=13 \ \\ a_{2}=3, b_{2}=8 \ \\ a_{3}=5, b_{3}=3 \ \\ \ \\ n=3

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