If the five digit number 235xy is divisible by 3,7 and 11, find value of (3x-4y)?

To solve the problem, we are given the five-digit number \( 235xy \), and it is divisible by 3, 7, and 11. This means that the number is divisible by the least common multiple (LCM) of 3, 7, and 11. Let's break down the solution step-by-step.

### Step 1: Find the LCM of 3, 7, and 11
Since 3, 7, and 11 are all prime numbers, their LCM is simply their product:
\[
\text{LCM}(3, 7, 11) = 3 \times 7 \times 11 = 231
\]
This means that \( 235xy \) must be divisible by 231.

### Step 2: Apply the divisibility rule for 3
A number is divisible by 3 if the sum of its digits is divisible by 3. The sum of the digits of \( 235xy \) is:
\[
2 + 3 + 5 + x + y = 10 + x + y
\]
For divisibility by 3:
\[
10 + x + y \equiv 0 \pmod{3}
\]
Thus, \( x + y \equiv -1 \equiv 2 \pmod{3} \). So:
\[
x + y \equiv 2 \pmod{3} \quad \text{(equation 1)}
\]

### Step 3: Apply the divisibility rule for 7 and 11
We now check divisibility by 7 and 11, but rather than applying their complex individual rules, we will check multiples of 231 close to 23500 to see which matches \( 235xy \).

The closest multiple of 231 near 23500 is:
\[
23500 \div 231 \approx 101.73 \quad \Rightarrow \quad 231 \times 102 = 23562
\]
So, the number \( 235xy = 23562 \), which implies \( x = 6 \) and \( y = 2 \).

### Step 4: Find \( 3x - 4y \)
Now that we know \( x = 6 \) and \( y = 2 \), we can calculate \( 3x - 4y \):
\[
3x - 4y = 3(6) - 4(2) = 18 - 8 = 10
\]

Thus, the value of \( 3x - 4y \) is \( \boxed{10} \).

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How do you find the LCM of more than two numbers?
2. What are the divisibility rules for different prime numbers?
3. How do you verify divisibility of larger numbers by smaller numbers?
4. What is the significance of modular arithmetic in divisibility problems?
5. Can you solve for \( 3x - 4y \) if a different condition is given?

**Tip**: Always start by finding the least common multiple (LCM) when solving divisibility problems involving multiple numbers.

If the five digit number 235xy is divisible by 3, 7 and 11, find value of (3x-4y)?

Solve the problem of finding (3x - 4y) when the number 235xy is divisible by 3, 7, and 11. This solution explores LCM, divisibility rules, and modular arithmetic to reach the answer. Perfect for students in Grades 9-11.

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