If a number 2x64y is complete divisible by 88, then what is the value of 6x – 5y

To determine the value of 6x - 5y, we need to find the values of x and y that make the number 2x64y divisible by 88.

Step 1: Divisibility by 8 For a number to be divisible by 8, the last three digits must form a multiple of 8. In this case, the last three digits are 64y. To find the possible values of y, we need to check which numbers, when multiplied by 64, result in a multiple of 8. The multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, and so on. From this list, we can see that y can be 1, 2, 3, 4, 5, 6, 7, 8, 9, or 0.

Step 2: Divisibility by 11 For a number to be divisible by 11, the difference between the sum of the digits in odd positions and the sum of the digits in even positions must be a multiple of 11. In this case, the number is 2x64y. The sum of the digits in odd positions is 2 + 6 + y = 8 + y, and the sum of the digits in even positions is x + 4. So, the difference is (8 + y) - (x + 4) = 4 + y - x.

To find the possible values of x and y that make the number divisible by 11, we need to find the values of x and y that make 4 + y - x a multiple of 11. The possible values for 4 + y - x are -7, 4, 15, 26, 37, 48, 59, 70, 81, 92, and so on. From this list, we can see that the possible values for 4 + y - x are 15, 26, 37, 48, 59, 70, 81, 92, and so on.

Step 3: Finding the values of x and y To find the values of x and y, we need to find the intersection of the possible values of y from step 1 and the possible values of 4 + y - x from step 2. From step 1, we know that y can be 1, 2, 3, 4, 5, 6, 7, 8, 9, or 0. From step 2, we know that the possible values for 4 + y - x are 15, 26, 37, 48, 59, 70, 81, 92, and so on.

By comparing the two lists, we can see that the only possible value for y that satisfies both conditions is y = 6. Plugging this value into the equation 4 + y - x = 15, we can solve for x:

4 + 6 - x = 15 10 - x = 15 -x = 5 x = -5

Step 4: Finding the value of 6x - 5y Now that we have found the values of x and y, we can substitute them into the equation 6x - 5y:

6(-5) - 5(6) -30 - 30 -60

Therefore, the value of 6x - 5y is -60.