Question

Assistance in understanding and solving this example on Elementary Number Theory with the details of the solution to better understand, thanks.

a) Find all solutions to 3x+4y=60 in positive integers.

b) A roadside stand bought 11 large baskets of eggs from a farmer and sold 39 small baskets of eggs, which hold fewer than a dozen. There were 19 eggs left over. How many eggs does a large basket hold?

Answer 1

a)

x | y

==========

16 | 3

12 | 6

8 | 9

4 | 12

b) A large basket contains 23 eggs

Explanation:

a) Isolating 3x, we get

3x = 60-4y

This means that 60-4y is a multiple of 3.

Besides, as x is a positive integer, then x = 60-4y has to be greater than zero

60-4y>0

Solving the inequality for y

60>4y

60/4 > y

15>y (or y<15, which is the same)

The multiples of 3 which are smaller than 15 are 3,6,9 and 12

Now we draw a table for these values of y and found the value of x by replacing in the equation 3x = 60-4y---> x = (60-4y)/3

x | y

=======

16 | 3

12 | 6

8 | 9

4 | 12

And these are all the solutions in positive integers.

b) This problem is pretty much like problem a)

Let's call x the number of eggs contained in the large baskets and y the number of eggs contained in the small baskets.

Then

Number of eggs bought - number of eggs sold = number of eggs left over.

In this case

11x-39y = 19

But we also know that 0<y<12 because the small baskets hols fewer than a dozen.

Isolating x in the equation 11x-39y + 19, we get

x=(19+39y)/11

Now we make a table with these values of x and y remembering that y is a positive integer smaller than 12

x | y

=======

5.2 | 1

8.8 | 2

12.4 | 3

15.9 | 4

19.5 | 5

23 | 6

26.6 | 7

30.1 | 8

33.6 | 9

37.2 |10

40.7 |11

From this table we see that the only integer solution for x is 23, so a large basket contains 23 eggs.