What is the largest two-digit number that is divisible by both of its digits, and the digits are distinct?


To find the largest two-digit number that is divisible by both of its digits, with distinct digits, follow these steps:

1. Let the two-digit number be \( ab \), where \( a \) and \( b \) are the digits of the number \( 10a + b \), and \( a \neq b \).
2. The number \( 10a + b \) must be divisible by both \( a \) and \( b \).
3. We need to check each two-digit number starting from the largest (99) down to the smallest (10) to find the number that meets these criteria.

### Step-by-step Process:

1. Start from 98 and go downwards, checking if the number is divisible by both of its digits.
2. To check a number \( 10a + b \):
   - Ensure \( (10a + b) \mod a = 0 \) (divisibility by the first digit)
   - Ensure \( (10a + b) \mod b = 0 \) (divisibility by the second digit)

### Checking Each Number:
Let's break down the numbers systematically:
- **96**: \( 9 \neq 6 \). \( 96 \div 9 = 10.67 \) (Not divisible by 9)
- **95**: \( 9 \neq 5 \). \( 95 \div 9 = 10.56 \) (Not divisible by 9)
- **94**: \( 9 \neq 4 \). \( 94 \div 9 = 10.44 \) (Not divisible by 9)
- **93**: \( 9 \neq 3 \). \( 93 \div 9 = 10.33 \) (Divisible by 9), \( 93 \div 3 = 31 \) (Divisible by 3)
  - \( 93 \) satisfies the conditions but let's keep going to find larger numbers.

Continue this process until reaching a larger number that fulfills the conditions. Upon checking systematically:

- **84**: \( 8 \neq 4 \). \( 84 \div 8 = 10.5 \) (Not divisible by 8)
- **81**: \( 8 \neq 1 \). \( 81 \div 8 = 10.125 \) (Not divisible by 8)
- **72**: \( 7 \neq 2 \). \( 72 \div 7 = 10.2857 \) (Not divisible by 7)
- **66**: Not distinct digits.

### The Correct Number:
After this process, we find:

- **48**: \( 4 \neq 8 \). \( 48 \div 4 = 12 \) (Divisible by 4), \( 48 \div 8 = 6 \) (Divisible by 8)

The largest number that meets the conditions is **48**.

### Answer:
The largest two-digit number that is divisible by both of its distinct digits is **48**.

Would you like more details or have any questions?

1. What is the significance of a number being divisible by its digits?
2. How can divisibility rules help solve such problems quickly?
3. Are there similar problems with different constraints?
4. What happens if we consider three-digit numbers instead?
5. How would the solution change if the digits could be the same?

**Tip:** When solving divisibility problems, it can be helpful to use the properties and rules of numbers to narrow down the possibilities efficiently.

What is the largest two-digit number that is divisible by both of its digits, and the digits are distinct?

Discover the largest two-digit number divisible by both of its distinct digits. This problem explores number theory and divisibility rules through a detailed step-by-step solution. Perfect for students in Grades 6-8.

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