Lesson Nice entertainment problems related to divisibility property

Nice entertainment problems related to divisibility property

Problem 1

Let  x  and  y  be two positive integers John thinks about.


Then from the condition, you have this equation

    xy - x - y = 35.


Add  1  (one)  to both side. You will get

    xy - x - y + 1 = 36.


It is equivalent to

    (x-1)*(y-1) = 36


Thus the integer number 36 is presented as the product of integer positive factors  (x-1)  and (y-1).


The possible factors of the number 36 are

    1, 2, 3, 4, 6, 9, 12, 18, 36.


So, (x-1) may have these values  1, 2, 3, 4, 6, 9, 12, 18, 36

(and, correspondingly, (y-1) has the values  of the same list in reversed order).


So, all possible values for numbers x and y are these pairs

    (x,y) = (2,37), (3,19), (4,13), (7,7), (10,5), (13,4), (19,3) and (37,2).      ANSWER

Problem 2

Let x be the number of baskets collected by some girl, and y be the number of basket collected by her brother.


Then for each such a pair, we have this equation

    x^2 - y^2 = 45.


Factor left side to get

    (x-y)*(x+y) = 45.


45 has these decompositions in the product of factors  1*45,  3*15  and  9*5.


These decompositions produce these systems of equations


a)  x - y =  1
    x + y = 45

        with the solution  x = 23, y = 22;


b)  x - y =  3
    x + y = 15

        with the solution  x = 9, y = 6;


c)  x - y =  5
    x + y =  9

        with the solution  x = 7, y = 2.



These are all possible solutions to the numbers of baskets collected by the children.

There is NO other solutions.



Hence, the total pay was   = 1183 dollars.    ANSWER