121 points] Always explain and justify all of your steps and answers. (a) Does t

Question

121 points] Always explain and justify all of your steps and answers. (a) Does the diophantine equation 1. have any integer solutions? If yes, find a solution. (b) Does the diophantine equation have any integer solutions other than the trivial solution z = 0,z = 0? If yes, find such a solution. (c) For any of the previous parts, if there is a solution, give a one-parameter family of solutions. (d) The age of a mother is one less than twice the age of her daughter, and the digits ay of the mother's age are reversed in the daughter's age. How old are they? [Hint: The number written in digits as ry is 10r +y]

Explanation / Answer

(a) GIVEN DIOPHANTINE EQUAT.ION HAS NO SOLUTION.

X2- 5Z =3 ,G.C.D OF COEFFICIENT OF LEFT SIDE = 1

DIVIDING BOTH SIDE BY 1 WE GET X2 - 5 Z = 3, NOW WE TRY TO SOLVE IT USING 9,16,25 AND FOUND NO SOLUTION. HENCE NO SOLUTION

(b) GIVEN EQUATION X2 - 5 Y = 0 , G.C.D. LEFT SIDE COEFFICIENT (1.-5) = 1

DIVIDING BOTH SIDE BY 1 EQUATION X2- 5 Z = 0

NOW USING 9.16, 25 TIOSWE FIND SOLUTION USING 25 HENCE PROCEED

NOW FROM EQUATION X2 = 5 Y ,HENCE` X` MUST BE A MULTIPLE OF `5

LET X = 5 N (WHERE `N` IS AN INTEGER) , ( 5 N )2 = 5 Y OR 25 N2= 5Y

OR Y = 5 N2 , THUS X = 5 N, Y = 5 N2 ARE SOLUTION

PUTTING N = 1,2,3,............ SOLUTIONS ARE ( 5,5 ), (10,20), (15.45) ..............AND SO ON.

(c) LET MOTHER`S AGE 10 X + Y THEN DAUGHTER`S AGE WILL BE GIVEN AS 10 Y + X AS DIGITS IN MOTHER`S AS `XY` IS REVERSED IN DAUGHTER`S AGE

NOW GIVEN THAT MOTHER IS ONE LESS THAN DOUBLE OF DAUGHTER`S AGE

HENCE 10 X+ Y = 2 (10 Y + X) -1 OR 10 X +Y-20 Y-2X = -1 OR 8X - 19 Y = -1

OR -8X + 19 Y = 1, G.C.D. OF COEFFICIENT OF LEFT SIDE (- 8 , 19) = 1, DIVIDING BOTH SIDE BY 1

EQUATION IS -8X + 19 Y = 1

FINDING G.C.D. OF -8 AND 19 WE PROCEED AS

19 = 2 (8) + 3 [ 3 = 19 - 2 (8) ]

8 = 2(3) + 2 [ 2 = 8 - 2 (3) ]

3 = 1 (2) + 1 [ 1 = 3 - 1 (2)

NOW WE EXPRESS 1 ( RIGHT SIDE VALUE OF EQUATION) IN LINEAR COMBINATION OF - 8, 19 AS

1 = 3 - 1 (2)

= 3 - 1 [ 8 - 2 (3)] =3 - 8 + 2 (3)

= 3 (3) - 8 = 3 [ 19 - 2 (8) ] - 8

1 = 3 (19) - 6 (8) -8 = 3(19) - 7 (8) AND GIVEN EQUATION -8X + 19 Y =1

COMPARING X = 7 , Y =3 , MOTHER`S AGE = 10 X7 + 3 =73 AND DAUGHTER`S 37

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