To determine the greatest volume of tins that the shopkeeper needs to fill the 16 liters, 24 liters, and 32 liters of three kinds of oil equally, we need to find the greatest common divisor (GCD) of these three numbers. The GCD is the largest number that can exactly divide each of the given volumes without leaving any remainder.
Let's denote the volumes of the oils as:
- Oil 1: 16 liters
- Oil 2: 24 liters
- Oil 3: 32 liters
Step-by-step solution:
1. Factorization of 16:
- 16 can be expressed as [tex]\(2 \times 2 \times 2 \times 2\)[/tex] or [tex]\(2^4\)[/tex].
2. Factorization of 24:
- 24 can be expressed as [tex]\(2 \times 2 \times 2 \times 3\)[/tex] or [tex]\(2^3 \times 3\)[/tex].
3. Factorization of 32:
- 32 can be expressed as [tex]\(2 \times 2 \times 2 \times 2 \times 2\)[/tex] or [tex]\(2^5\)[/tex].
4. Find the common prime factors:
- The common prime factor among 16, 24, and 32 is 2.
5. Determine the lowest power of the common factor:
- In the factorization of 16, the power of 2 is 4 ([tex]\(2^4\)[/tex]).
- In the factorization of 24, the power of 2 is 3 ([tex]\(2^3\)[/tex]).
- In the factorization of 32, the power of 2 is 5 ([tex]\(2^5\)[/tex]).
The lowest power of 2 common to all three numbers is [tex]\(2^3\)[/tex].
6. Calculate the greatest common divisor:
- The GCD of 16, 24, and 32 is determined by the common factor [tex]\(2^3\)[/tex]. Since [tex]\(2^3 = 8\)[/tex], the greatest common divisor is 8.
Therefore, the greatest volume of tins the shopkeeper needs to fill the oils equally is 8 liters.
Thus, the solution to the problem is:
[tex]\[ \boxed{8} \][/tex]
