How many positive integers less than or equal to 2000 have an odd number of factors?
1 answer:
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Answer:
We have 4 solutions:
- No 10-foot truck, 10 14-foot trucks, and 15 24-foot trucks
- 2 10-foot trucks, 7 14-foot trucks, and 16 24-foot trucks
- 4 10-foot trucks, 4 14-foot trucks, and 17 24-foot trucks
- 6 10-foot trucks, 1 14-foot trucks, and 18 24-foot trucks
Step-by-step explanation:
Let the number of 10-foot truck with a capacity of 350 cubic feet purchased=a
Let the number of 14-foot truck with a capacity of 700 cubic feet purchased=b
Let the number of 24-foot truck with a capacity of 1,400 cubic feet purchased=c
The company wants to purchase 25 trucks, therefore.
- a+b+c=25
Furthermore, the combined capacity of the trucks is 28,000 cubic feet.
- 350a+700b+1400c=28000
Since the number of equations is less than the number of variables, you can not use a matrix equation to solve this problem. The solution is most easily found using an augmented matrix.
The augmented matrix is presented below:
Using the calculator, the reduced row echelon form is:
where
a- 2c=-30 means a =2c-30
b+3c=55 means b= 55-3c
We alter the value of c as long as neither a nor b becomes negative. Suitable values for c are 15, 16, 17, and 18:
We can easily verify that, for each solution, the number of trucks adds up to 25 and the fleet capacity is 28,000 cubic feet.
We therefore have 4 solutions:
- No 10-foot truck, 10 14-foot trucks, and 15 24-foot trucks
- 2 10-foot trucks, 7 14-foot trucks, and 16 24-foot trucks
- 4 10-foot trucks, 4 14-foot trucks, and 17 24-foot trucks
- 6 10-foot trucks, 1 14-foot trucks, and 18 24-foot trucks