If the number of positive integral solutions of the form (x, y) of the
equation 5x + 4y = 221 is equal to k, then the sum of digits of k will be
equal to
Answers
Answer:
Number of Integral Solutions) Step-3:If you understand the 4th point mentioned above, at one of any two consecutive integral values of y, the value of x will come out to be an integer OR at one of the 3 consecutive values of x, the value of y will come out to be an integer. That means, if we add 2n (where n is an integer) to the first value for y, we will have to subtract 3n from the first value of x to get integral solutions. That means,
If y =1 +2(1) = 3 , x= 18-3(1) = 15.
If y= 1 + 2(2) = 5, x= 18 – 3(2) = 12.
If y= 1 + 2(3) = 7, x = 18 – 3(3) = 9 and so on.
(Number of Integral Solutions) Step-4:This equation will have infinite number of integral solutions but finite number of non-negative integral solutions. Let’s see how we can find it.
We can keep increasing the value of y in the positive direction but x will be decreasing simultaneously and become less than 0 at one point. As lowest non negative integral value of y is 1,highest allowable positive value of x is 18 and it is decreasing by 3. So, x can take 7 non negative integral values and they are- 18, 15, 12, 9, 6, 3 and 0.Hence the given equation has 7 non negative integral values.
Note: In equation Ax + By = C, if C is divisible by any of A or B, then number of non-negative integral solutions = {C/LCM(A,B)} + 1
Equation type: x1+x2+⋯+xr=n
Case-1: Positive integral solutions.
Let us understand the concept from an example:
X1 + X2 + X3= 8.
Answer:
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