Divisibility Rules

When a number is divisible by both 7 and 16, it will also be divisible by their common factors. To understand why, let's first discuss the divisibility rules for 7 and 16.

Divisibility Rule for 7:
To check if a number is divisible by 7, we can use the following rule:
- Take the last digit of the number and double it.
- Subtract this result from the remaining part of the number (excluding the last digit).
- If the result is divisible by 7, then the original number is also divisible by 7.

Divisibility Rule for 16:
To check if a number is divisible by 16, we can use the following rule:
- Check if the last four digits of the number form a multiple of 16.
- If the last four digits are a multiple of 16, then the original number is also divisible by 16.

Common Factors of 7 and 16:
Now, let's consider a number that is divisible by both 7 and 16. Since 7 and 16 are relatively prime (they have no common factors other than 1), the common factors of the two numbers are the factors of their product, which is 7 * 16 = 112.

So, a number that is divisible by both 7 and 16 will also be divisible by 112.

Explanation:
When a number is divisible by both 7 and 16, it means that it satisfies the divisibility rules of both numbers. This implies that the number can be written in the form of 7a and 16b, where a and b are integers.

Since the number is divisible by 7, it can be written as 7a. Similarly, since the number is divisible by 16, it can be written as 16b.

Now, we can express the number as a product of its prime factors:
7a = 2^3 * 7 * b.

From this expression, we can conclude that the number is divisible by both 2^3 (8) and 7. Therefore, it will also be divisible by their product, which is 8 * 7 = 56.

Furthermore, since 56 is a factor of both 8 and 7, it is also a factor of the original number. Hence, a number that is divisible by both 7 and 16 will also be divisible by 56.

In summary, a number that is divisible by both 7 and 16 will also be divisible by their common factors: 112 and 56.