=> x ,x + 4 thuộc Ư(20) = {1;2;4;5;10;20}

To be able to buy 9 bottles of fruit juice at the cheapest price, they should appoint 4 people to buy 8 bottles for: (7 + 1) x 4 = 32 (dollars)

The total amount they will pay to buy 9 bottles is: 32 + 7 = 39 (dollars)

Give : 

Change x,y,z into 5z - 3x - 4y = 50 , we have :

5(6k + 5) - 3(2k + 1) - 4(4k - 3) = 50

30k + 25 - 6k - 3 - 16k + 12 = 50 

8k + 34 = 50

8k = 16

=> k = 2

At the begining, there are 1974/2=987 odd numbers. When we replace two numbers a and b by a number either a + b or a - b, we see that:

+ if a odd and b even, or a even and b odd then a + b or a - b is still odd

+ if a and b are both even then a + b or a -b is still even

+ If a and  are both odd then a + b or a - b is even

So the number of odd after each such replacement operation is stay the same or deacresed by two. At the begining, there is 987 odd numbers  (987 is odd) and the odd number left must be odd too. So at the final, only one number left, it must be odd number, it is not 0. (because 0 is even).

1339 = 103 x 13 = 1 x 1339.

No one can be 1339 years old, so the grandfather will be 103 years old and his grandchild will be 13 years old next year.

Thus, the grandfather is 102 years old, and his grandchild is 12 years old now.