To determine which factors of 72 are multiples of its prime factors, we first need to find the prime factorization of the number 72.
The prime factorization of 72 is accomplished by dividing it by the smallest prime numbers:
- 72 ÷ 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
This gives us the prime factorization of 72 as:
72 = 2^3 × 3^2
From this factorization, we identify the prime factors as 2 and 3.
Now, let’s list out all the factors of 72:
- 1
- 2
- 3
- 4
- 6
- 8
- 9
- 12
- 18
- 24
- 36
- 72
Next, we identify which of these factors are multiples of the prime factors (2 and 3).
- 2 (2 × 1)
- 4 (2 × 2)
- 6 (2 × 3)
- 8 (2 × 4)
- 12 (2 × 6)
- 18 (2 × 9)
- 24 (2 × 12)
- 36 (2 × 18)
- 72 (2 × 36)
For multiples of 3, we have:
- 3 (3 × 1)
- 6 (3 × 2)
- 9 (3 × 3)
- 12 (3 × 4)
- 18 (3 × 6)
- 36 (3 × 12)
- 72 (3 × 24)
Combining these lists, the factors of 72 that are multiples of its prime factors, 2 and 3, are:
- 2
- 3
- 4
- 6
- 8
- 9
- 12
- 18
- 24
- 36
- 72
In summary, all the factors of 72 listed above are indeed multiples of its prime factors.