Explain why Proposition 71 on p 126 guarantees that there ex

Explain why Proposition 7.1 on p. 126 guarantees that there exist integers x and y such that 97x + 20y = 1. Then find a pair of integers x and y that satisfy this condition. Is this pair of integers unique? In the country named Halfland the local currency is called Div, and the central bank prints only two kinds of notes: 5 Div and 11 Div. Prove that it is possible to pay any amount using these notes (assuming that the recipient of the payment can give any change needed). Would it still be possible to pay any amount if the notes were 4 Div and 6 Div?

Solution

a)

97x+20y=1

the pair of integers that satisfies this condition

here we get the result as positive number so choose y is negative number

here y coefficient is 20 any number multiplied by 20 is gives even and last digit is zero

so choose x such that 97 multiplied by x gives one as last digit

97 *3=301

choose 20y =1-301=-300

y=-15

x=3 and y=-15

it should not be unique

when we take x=23 then y=-112

b)the central bank prints only two kinds of notes 5 Div ,11Div

it is possible to pay any change when

i need 1 Div as change if i can pay 11 Div then in return i get two 5 Divs

i need 2 Div as change if i can pay 22 Div (two 11 Divs) then in return i get four 5 Divs

i need 3 Div as change if i can pay 33 Div (three 11 Divs) then in return i get six 5 Divs  

i need 4 Div as change if i can pay 44 Div (four 11 Divs) then in return i get eight 5 Divs

for some integers the difference of 11 multiples and 5 multiples will give 1,2,3,4 respectively so we get change needed

similarly for 6Div ,7Div, 8Div,9 Div when subtracted by 5 they will become 1,2,3,4 Div respectively as per above we described we can pay the amount

large amount greater than 10 Div will be divide by 5 and remaining given by using 11 Div

from this we can say that we can pay any amount using these 5Div, 11Div

c)

it can\'t possible to pay any amount if notes were 4 Div and 6 Div

i need 1 Div as change if i pay 4 Div    i can\'t get 3 Div or if i pay 6 Div there no note which is 5 Div

since 6 is multiple of 3 there we didn\'t get any number which the differnce of 3 multiples and 6 multiples gives 1,2

so we can\'t able get needed change like 1Div and 2Div