(a) List all the divisors (factors) of 48 . (b) List all the divisors (factors) of 54 . (c) Use parts (a) and (b) to find the largest common divisor of 48 and 54 . (a) The divisors of 48 are $$ \square $$. (Simplify your answer. Use a comma to separate answers as needed.)

Question

UpStudy AI · Accepted Answer

Check the equality by following steps:
- step0: Check the equality is true or false:
  $$48=2^{4}\times 3^{1}$$
- step1: Simplify:
  $$48=48$$
- step2: Check the equality:
  $$\textrm{true}$$
Solve the equation $$ 54=2^1*3^3 $$.
Check the equality by following steps:
- step0: Check the equality is true or false:
  $$54=2^{1}\times 3^{3}$$
- step1: Simplify:
  $$54=54$$
- step2: Check the equality:
  $$\textrm{true}$$
To find the divisors of 48 and 54, we can start by determining their prime factorizations.

### (a) Divisors of 48
The prime factorization of 48 is:
$$
48 = 2^4 \times 3^1
$$
To find the divisors, we can take all combinations of these prime factors. The divisors of 48 are:
1. $$2^0 \times 3^0 = 1$$
2. $$2^1 \times 3^0 = 2$$
3. $$2^2 \times 3^0 = 4$$
4. $$2^3 \times 3^0 = 8$$
5. $$2^4 \times 3^0 = 16$$
6. $$2^0 \times 3^1 = 3$$
7. $$2^1 \times 3^1 = 6$$
8. $$2^2 \times 3^1 = 12$$
9. $$2^3 \times 3^1 = 24$$
10. $$2^4 \times 3^1 = 48$$

Thus, the divisors of 48 are:
$$
1, 2, 3, 4, 6, 8, 12, 16, 24, 48
$$

### (b) Divisors of 54
The prime factorization of 54 is:
$$
54 = 2^1 \times 3^3
$$
Similarly, we can find the divisors:
1. $$2^0 \times 3^0 = 1$$
2. $$2^1 \times 3^0 = 2$$
3. $$2^0 \times 3^1 = 3$$
4. $$2^1 \times 3^1 = 6$$
5. $$2^0 \times 3^2 = 9$$
6. $$2^1 \times 3^2 = 18$$
7. $$2^0 \times 3^3 = 27$$
8. $$2^1 \times 3^3 = 54$$

Thus, the divisors of 54 are:
$$
1, 2, 3, 6, 9, 18, 27, 54
$$

### (c) Largest Common Divisor
Now, we can find the common divisors of 48 and 54:
- Common divisors: $$1, 2, 3, 6$$

The largest common divisor is:
$$
6
$$

### Summary
(a) The divisors of 48 are $$ 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 $$.

(b) The divisors of 54 are $$ 1, 2, 3, 6, 9, 18, 27, 54 $$.

(c) The largest common divisor of 48 and 54 is $$ 6 $$.
(a) The divisors of 48 are $$1, 2, 3, 4, 6, 8, 12, 16, 24, 48$$. (b) The divisors of 54 are $$1, 2, 3, 6, 9, 18, 27, 54$$. (c) The largest common divisor of 48 and 54 is $$6$$.

(a) List all the divisors (factors) of 48 . (b) List all the divisors (factors) of 54 . (c) Use parts (a) and (b) to find the largest common divisor of 48 and 54 . (a) The divisors of 48 are \( \square \). (Simplify your answer. Use a comma to separate answers as needed.)

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