Explanations:
To understand why a number divisible by 9 is also divisible by 3, we need to delve into the concept of divisibility and factors. A number is divisible by another number if it can be divided by that number without leaving a remainder. The divisibility rules are shortcuts to determine whether a number is divisible by another number without performing the actual division.
In the case of divisibility by 9, a number is divisible by 9 if the sum of its digits is divisible by 9. For example, the number 189 is divisible by 9 because the sum of its digits (1 + 8 + 9 = 18) is divisible by 9. This rule is derived from the fact that 9 is a multiple of 3, and the sum of the digits of a number can be used to determine divisibility by 9.
Now, let's consider the factors of 9. The factors of 9 are 1, 3, and 9. This means that any number divisible by 9 must also be divisible by its factors, which include 3. Therefore, if a number is divisible by 9, it must also be divisible by 3. This is because 9 can be factored into 3 × 3, and any number that is a multiple of 9 must also be a multiple of 3.
To further illustrate this, consider the number 18. It is divisible by 9 because 18 ÷ 9 = 2, and it is also divisible by 3 because 18 ÷ 3 = 6. This example demonstrates that any number divisible by 9 is also divisible by 3.
It is important to note that the other options provided (6 and 11) are not necessarily factors of a number divisible by 9. For a number to be divisible by 6, it must be divisible by both 2 and 3, but 9 does not guarantee divisibility by 2. Similarly, the divisibility rule for 11 is different and does not relate to the divisibility rule for 9.
In conclusion, the correct answer is that a number divisible by 9 is also divisible by 3. This is based on the fact that 9 is a multiple of 3, and any number divisible by 9 must also be divisible by its factors, which include 3.