Question

The number of positive integral solutions to the equation 4x + 5y + 2z = 111 is​

Answer 1

The number of positive integral solutions to the equation 4x + 5y + 2z = 111 is​

Step-by-step explanation:

The “y” must have odd values: 1, 3, 5 …..... 21

For y = 23: then 5*y would be 115

∴, there are 11 values for “y”.

We start with the highest, the 21.

5*y = 5*21 = 105

∴: 4x+105+2z = 111

∴: 4x + 2z = 6

∴: 2x + z = 3

One solution: x=1; z=1; y = 21

For y=19

5*y = 5*19 = 95

∴: 4x+95+2z = 111

∴: 4x + 2z = 16

2x + z = 8

Hence, z must be even and less than 8: 2, 4, 6 …

z = 2; x=3; y =19

z = 4; x=2; y = 19

z = 6; x=1; y = 19

For y = 17

5*y = 85

∴: 4x + 2z = 26

∴ 2x + z = 13

Then z must be odd, from 1 to 9: 1, 3, 5, 7, 9

Those are 5 solutions.

For every y the number of solutions is odd in arithmetic progression as y decreases.

the number of solutions will be:

1+3+5+7+9+11+13+15+17+19+21 = 121