Which group shows all the whole answer number factors of a composite number?
A. 1,2,3,4,5,8,16
B. 1,2,3,4,5,6,10,20
C. 1,2,3,4,6,8,12,24
D. 1,2,3,4,7,14,28
A complete list of integer factors of a composite, sorted in ascending order will have the property that:
The first number multiplied by the last number equals the composite number,
the second number multiplied by the second last will also equal the composite number,
....
and so on.
If there is an even number of factors, the product of the two middle numbers will also equal the composite number.
If there is an odd number of factors, the composite number is a perfect square, with the middle number being the square-root of the number.
Based on these facts, try to find the answer from the list.
To find the group that shows all the whole number factors of a composite number, we need to list down all the factors of the given composite number.
A composite number is a positive integer that has more than two distinct positive divisors. In other words, it can be divided evenly by at least three different positive integers.
Let's analyze the options:
A. 1, 2, 3, 4, 5, 8, 16
B. 1, 2, 3, 4, 5, 6, 10, 20
C. 1, 2, 3, 4, 6, 8, 12, 24
D. 1, 2, 3, 4, 7, 14, 28
Now let's find the factors of the composite number by dividing it with different integers:
For Option A:
- If the composite number is divisible by 1: Yes (1 is a factor)
- If the composite number is divisible by 2: Yes (2 is a factor)
- If the composite number is divisible by 3: Yes (3 is a factor)
- If the composite number is divisible by 4: Yes (4 is a factor)
- If the composite number is divisible by 5: Yes (5 is a factor)
- If the composite number is divisible by 8: Yes (8 is a factor)
- If the composite number is divisible by 16: Yes (16 is a factor)
All the factors in option A are included in the list of factors of the composite number.
For Option B:
- If the composite number is divisible by 1: Yes (1 is a factor)
- If the composite number is divisible by 2: Yes (2 is a factor)
- If the composite number is divisible by 3: Yes (3 is a factor)
- If the composite number is divisible by 4: Yes (4 is a factor)
- If the composite number is divisible by 5: Yes (5 is a factor)
- If the composite number is divisible by 6: Yes (6 is a factor)
- If the composite number is divisible by 10: No (Not a factor)
- If the composite number is divisible by 20: No (Not a factor)
Options B does not include all the factors of the composite number.
For Option C:
- If the composite number is divisible by 1: Yes (1 is a factor)
- If the composite number is divisible by 2: Yes (2 is a factor)
- If the composite number is divisible by 3: Yes (3 is a factor)
- If the composite number is divisible by 4: Yes (4 is a factor)
- If the composite number is divisible by 6: Yes (6 is a factor)
- If the composite number is divisible by 8: Yes (8 is a factor)
- If the composite number is divisible by 12: No (Not a factor)
- If the composite number is divisible by 24: No (Not a factor)
Options C does not include all the factors of the composite number.
For Option D:
- If the composite number is divisible by 1: Yes (1 is a factor)
- If the composite number is divisible by 2: Yes (2 is a factor)
- If the composite number is divisible by 3: Yes (3 is a factor)
- If the composite number is divisible by 4: Yes (4 is a factor)
- If the composite number is divisible by 7: No (Not a factor)
- If the composite number is divisible by 14: No (Not a factor)
- If the composite number is divisible by 28: No (Not a factor)
Options D does not include all the factors of the composite number.
Therefore, the correct group that shows all the whole number factors of the composite number is:
A. 1, 2, 3, 4, 5, 8, 16