Understanding Divisibility by 88

A number is divisible by 88 if and only if it is divisible by both 8 and 11, because 8 and 11 are coprime factors of 88.

The given 6-digit number is $5x423y$. We need to find the values of the digits $x$ and $y$ based on the divisibility rules for 8 and 11.

Applying the Divisibility Rule for 8

A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

For the number $5x423y$, the last three digits form the number $23y$. Therefore, $23y$ must be divisible by 8.

Let's test possible values for $y$ (which must be a digit from 0 to 9):

• If $y=0$, $230 \div 8$ leaves a remainder.
• If $y=1$, $231 \div 8$ leaves a remainder.
• If $y=2$, $232 \div 8 = 29$. This is exactly divisible.
• If $y=3$, $233 \div 8$ leaves a remainder.
• If $y=4$, $234 \div 8$ leaves a remainder.
• If $y=5$, $235 \div 8$ leaves a remainder.
• If $y=6$, $236 \div 8$ leaves a remainder.
• If $y=7$, $237 \div 8$ leaves a remainder.
• If $y=8$, $238 \div 8$ leaves a remainder.
• If $y=9$, $239 \div 8$ leaves a remainder.

The only single digit $y$ for which $23y$ is divisible by 8 is $y=2$. So, we know that $y=2$.

Applying the Divisibility Rule for 11

A number is divisible by 11 if the alternating sum of its digits is divisible by 11.

For the number $5x423y$, the alternating sum of the digits is calculated as:

$( \text{Sum of digits at odd places from the right} ) - ( \text{Sum of digits at even places from the right} )$

The digits from right to left are $y, 3, 2, 4, x, 5$.

Alternating sum $= y - 3 + 2 - 4 + x - 5$

We know $y=2$, so substitute $y=2$ into the expression:

Alternating sum $= 2 - 3 + 2 - 4 + x - 5$

Alternating sum $= (2 + 2 + x) - (3 + 4 + 5)$

Alternating sum $= (4 + x) - 12$

Alternating sum $= x - 8$

According to the divisibility rule for 11, this alternating sum $(x - 8)$ must be divisible by 11. Since $x$ is a single digit (0-9), the possible values for $(x-8)$ that are divisible by 11 are $0, 11, -11, 22, -22, \dots$.

Let's check the possibilities for $(x-8)$ within the range of possible values for $x-8$ when $x \in \{0, 1, \dots, 9\}$ (i.e., $x-8 \in \{-8, -7, \dots, 1\}$):

• If $x - 8 = 0$, then $x = 8$. This is a valid digit (0-9).
• If $x - 8 = 11$, then $x = 19$. Not a valid digit.
• If $x - 8 = -11$, then $x = -3$. Not a valid digit.
• Any other multiple of 11 would result in $x$ being outside the range 0-9.

Therefore, the only possible value for $x$ is 8. So, we know that $x=8$.

Calculating the Value of (5x - 8y)

We have found that $x=8$ and $y=2$. Now we need to find the value of the expression $(5x - 8y)$.

Substitute the values of $x$ and $y$ into the expression:

$5x - 8y = 5(8) - 8(2)$

$5x - 8y = 40 - 16$

$5x - 8y = 24$

The value of $(5x - 8y)$ is 24.

Revision Table: Divisibility Rules Applied

Additional Information: Importance of Divisibility Rules

Divisibility rules are useful shortcuts for determining if a number is divisible by another number without performing long division. Understanding these rules can significantly speed up calculations in various mathematical problems and tests.

• Divisibility by 8: Check the number formed by the last three digits. If this number is divisible by 8, the original number is also divisible by 8.
• Divisibility by 11: Calculate the alternating sum of the digits (starting from the rightmost digit, subtract the next, add the next, and so on). If the result is 0 or a multiple of 11, the original number is divisible by 11.

When a number must be divisible by a composite number like 88 (which is $8 \times 11$), and the factors (8 and 11) are coprime (their greatest common divisor is 1), the number must be divisible by each of these coprime factors individually. This is a fundamental concept used in solving this type of problem.