indeterminate equation, JMO

Students of a class are made to stand in rows. If 4 students are extra in each 

row, there would be 2 less rows. If 4 students are less in each row, there would 

be 4 more rows. Find the total number of students which satisfies this condition 

if the number of students is less than 100.




Answer:

let number of students in a row be "x"

let number of rows be "y"

total number of students = xy

given, (x + 4)(y - 2) = xy ----------(1)

and, (x - 4)(y + 4) = xy   -----------(2)

So, (x + 4)/(x - 4) = (y + 4)/(y - 2) equating (1) and (2)

{x + 4 + x - 4}/{x + 4 - x + 4} = {y + 4 + y - 2}/{y + 4 - y + 2}

using componendo and dividendo

therefore, 2x/{8} = {2y + 2}/{6}

or x/4 = {y + 1}/3

or x = (4/3) {y + 1}

Since x is a whole number,

So, (y + 1)  should be a multiple of 3 or = 3n where n is an element of natural 

numbers. So, y = 3n - 1

so, possible values of y are,   2, 5, 8, 11, 14, 17 ...............

for which the values of x are  4, 8,12, 16, 20, 24 ...............

therefore, value of xy are      8, 40, 96, 176, ..........................

Therefore, there is only 1 possible value of x,y which satisfies the conditions 

(x,y) = (2,4) and (5,8) does not satisfy equation (1) and (2)

The total number of students is 96