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Question

Considering all 2-digit natural numbers, how many values of "y" do not satisfy the equation |7x-5y| = 3, given that "x"and "y" are positive integers.

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Solution


|7x5y|=3;7x5y=±3;
7x35=y;(A)
7x35=y;(B)
Case A : 7x35=y;(A)
x,y 2-digit natural numbers 10y99 and 10x99
7x = 498 x = 71 max, integral value
7x = 53 x = 8 min, integral value
Thus for 10y99x lies between 8 and 71.
But x should lie between 10 and 99,
For that y should be greater than 12
Hence 12y99.
For y = 12; x = 9 value not possible.
Since, 7x=5×y+3x=5×y+37;
we should check next value of y at 19. [12+7 = 19].
Values of y will be 19, 26, 33, ...., 96.
96=19+(n1)×7n=12 number of values of y that satisfy the equation.

Case B : 7x35=y;(B)
x,y 2-digit natural numbers 10y99;10x99
7x = 492 x = 70 max, integral value
7x = 47 x = 7 min, integral value
Thus for 10y99x lies between 7 and 70.
But x should lie between 10 and 99,
For that y should be greater than 15
Hence 15y99.
For y=16;x=11
Since, 7x=5×y3x=5×y37;
we should check next value of y at 23. [16+7 = 23].
x=5×2337=16
Values of y will be 16,23,30, ...., 93.
93=16+(n1)7n=12 number of values of y that satisfy the equation.
Total number of values = n(case A) + n(case B)=12+12=24
Total number of values of y between 10 and 99,
99=10+(n-1)1 n=90
No. of values of y that satisfy equation = 24
Total no. of values = 90
Hence, no. of values that do not satisfy the equation = 99-24=66.

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