Understanding Divisibility and the Problem

The question asks us to find the value of the expression $(3x - 5y)$ for a specific 5-digit number, 676xy. This number has the form 676 followed by two digits, x and y. The key condition is that this number must be divisible by 3, 7, and 11 simultaneously.

When a number is divisible by multiple numbers, it must also be divisible by the Least Common Multiple (LCM) of those numbers. In this case, the numbers are 3, 7, and 11.

Calculating the Least Common Multiple (LCM)

To find the number 676xy, we first need to find the LCM of 3, 7, and 11.

The numbers 3, 7, and 11 are all prime numbers. The LCM of distinct prime numbers is simply their product.

LCM$(3, 7, 11) = 3 \times 7 \times 11 = 21 \times 11 = 231$.

So, the 5-digit number 676xy must be divisible by 231.

Finding the Specific 5-Digit Number 676xy

The number is 676xy, which means it is between 67600 (when x=0, y=0) and 67699 (when x=9, y=9).

We are looking for a multiple of 231 that falls within this range (67600 to 67699).

Let's divide a number close to the start of the range by 231 to find the approximate multiple.

Consider 67600:

$\frac{67600}{231} \approx 292.64$

This tells us that the multiple of 231 we are looking for will be around the 293rd multiple or slightly higher if 292nd is below 67600.

Let's calculate the 292nd and 293rd multiples of 231:

• $231 \times 292 = 67452$
• $231 \times 293 = 67683$

The number $67452$ is not in the form 676xy (it's in the form 674yz).

The number $67683$ is in the form 676xy. Here, the digits after 676 are 8 and 3.

So, the number is 67683.

Identifying the Values of x and y

Comparing 676xy with 67683, we can identify the values of x and y:

• $x = 8$
• $y = 3$

Verifying Divisibility (Optional Check)

Let's quickly check if 67683 is divisible by 3, 7, and 11 using divisibility rules:

• Divisibility by 3: Sum of digits = $6 + 7 + 6 + 8 + 3 = 30$. Since 30 is divisible by 3, 67683 is divisible by 3.
• Divisibility by 7: This rule is more complex. A direct division is often easier: $67683 \div 7 = 9669$. It is divisible by 7.
• Divisibility by 11: Alternating sum of digits starting from the right: $(3 - 8 + 6 - 7 + 6) = (3+6+6) - (8+7) = 15 - 15 = 0$. Since 0 is divisible by 11, 67683 is divisible by 11.

The number 67683 satisfies all the divisibility conditions.

Calculating the Value of (3x - 5y)

Now that we have $x=8$ and $y=3$, we can calculate the value of the expression $(3x - 5y)$.

Substitute the values of x and y into the expression:

$3x - 5y = 3(8) - 5(3)$

$3(8) = 24$

$5(3) = 15$

So, $3x - 5y = 24 - 15 = 9$.

The value of $(3x - 5y)$ is 9.

Revision Table: Key Concepts Reviewed

Additional Information: Number Theory and Divisibility

Number theory deals with the properties of integers. Divisibility rules are shortcuts to determine if one number is divisible by another without performing long division. Understanding LCM is crucial when a number needs to be divisible by several numbers simultaneously, as it tells you the smallest number that satisfies all individual divisibility conditions.

For this problem, finding the LCM first simplifies the search for the 5-digit number. Instead of checking divisibility by 3, 7, and 11 for many numbers, we only check divisibility by 231.

The range 676xy implies numbers from 67600 up to 67699. Identifying the multiples of 231 near the beginning of this range helps pinpoint the required number efficiently.