A number is always divisible by 90, if ______

Options

A. It is divisible by both 2 and 45
B. It is divisible by both 5 and 18
C. It is divisible by both 9 and 10
D. All of these

Answer

To determine when a number is always divisible by 90, we need to analyze the options provided in the question. Let's break down the solution step by step.

### Step 1: Understand the Divisibility Rule for 90
A number is divisible by 90 if it is divisible by both 9 and 10. This is because 90 can be factored into its prime components: \(90 = 9 \times 10\).

### Step 2: Check the Options
We have four options to check:
1. Divisible by both 2 and 45
2. Divisible by both 5 and 18
3. Divisible by both 9 and 10
4. All of these

### Step 3: Analyze Each Option
- **Option 1: Divisible by both 2 and 45**
- The LCM of 2 and 45 is 90 (since they are co-prime). Therefore, if a number is divisible by both 2 and 45, it is divisible by 90.

- **Option 2: Divisible by both 5 and 18**
- The LCM of 5 and 18 is also 90 (since they are co-prime). Thus, if a number is divisible by both 5 and 18, it is divisible by 90.

- **Option 3: Divisible by both 9 and 10**
- The LCM of 9 and 10 is 90. Hence, if a number is divisible by both 9 and 10, it is divisible by 90.

### Step 4: Conclusion
Since all three options lead to the conclusion that the number is divisible by 90, we can say that the answer is:
**A number is always divisible by 90 if it is divisible by both 2 and 45, or both 5 and 18, or both 9 and 10. Therefore, the correct answer is "All of these."**