If the 5-digit number 676xy is divisible by 3, 7 and 11, then what is the…

To solve this problem, we need to determine the values of x and y such that the 5-digit number 676xy is divisible by 3, 7, and 11.

Step-by-Step Calculation

1. Find the Divisor: A number divisible by 3, 7, and 11 must be divisible by their Least Common Multiple (LCM). Since 3, 7, and 11 are all prime numbers, their LCM is simply their product: LCM = 3 * 7 * 11 = 231.

2. Determine the Number: We are looking for a number in the form 676xy. The smallest possible value is 67600 and the largest is 67699. We can find the specific number by dividing 67600 by 231 to see how close it is to a multiple of 231.

   • Dividing 67600 by 231: 67600 = 231 * 292 + 148.

   • To find the next multiple of 231, we add the difference: 231 - 148 = 83.

   • So, 67600 + 83 = 67683.

   • This matches the 676xy format where x = 8 and y = 3.

3. Calculate the Expression: We need to find the value of (3x - 5y):

   • Substitute x = 8 and y = 3:

   • 3(8) - 5(3) = 24 - 15 = 9.