# Answer to Question #12982 in Abstract Algebra for Kakay Gascon

Question #12982

vicky is thinking a number a number less than 700.when she interchanges ones and hundreds digit, the number increases by 297. when she interchanges the ones and tens digit, the result increases by 18.can you find the number she thinking of?

Expert's answer

Let xyz be the literal presentation of riddled number. Then the following statements take place:&

xyz<700& (1)

zyx - xyz = 297& (2)

xzy - xyz = 18& (3)

Obviously,

xyz<zyx ==> x<z

xyz<xzy ==> yz<zy ==> y<z

xyz<700 ==> x<7

Let's consider (3). Let z be 9. Then

z = 9 ==> 9y = 18 + y9 = [y+2]7 => y = 7.

Really, 97-79=18.

97x - x79 = 297 ==> x = 6.

Really, 976 - 679 = 297, 697 - 679 = 18 and 679<700. So, riddled number is 679.

xyz<700& (1)

zyx - xyz = 297& (2)

xzy - xyz = 18& (3)

Obviously,

xyz<zyx ==> x<z

xyz<xzy ==> yz<zy ==> y<z

xyz<700 ==> x<7

Let's consider (3). Let z be 9. Then

z = 9 ==> 9y = 18 + y9 = [y+2]7 => y = 7.

Really, 97-79=18.

97x - x79 = 297 ==> x = 6.

Really, 976 - 679 = 297, 697 - 679 = 18 and 679<700. So, riddled number is 679.

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