Understanding Divisibility Rules for 88

A number is considered divisible by 88 if and only if it is divisible by both 8 and 11. This is because 8 and 11 are co-prime factors of 88 (their greatest common divisor is 1). We will use the divisibility rules for 8 and 11 to find the values of A and B in the number 8A5146B.

Applying the Divisibility Rule for 8

The divisibility rule for 8 states that a number is divisible by 8 if the number formed by its last three digits is divisible by 8.

In the number 8A5146B, the last three digits are 46B.

For 46B to be divisible by 8, we can test values for B from 0 to 9:

• If B = 0, 460 is not divisible by 8 (460 ÷ 8 = 57.5).
• If B = 1, 461 is not divisible by 8.
• If B = 2, 462 is not divisible by 8.
• If B = 3, 463 is not divisible by 8.
• If B = 4, 464 is divisible by 8 (464 ÷ 8 = 58).
• If B = 5, 465 is not divisible by 8.
• If B = 6, 466 is not divisible by 8.
• If B = 7, 467 is not divisible by 8.
• If B = 8, 468 is not divisible by 8.
• If B = 9, 469 is not divisible by 8.

The only value for B that makes 46B divisible by 8 is B = 4.

So, the number is 8A51464.

Applying the Divisibility Rule for 11

The divisibility rule for 11 states that the difference between the sum of the digits at the odd places (from the right) and the sum of the digits at the even places (from the right) must be either 0 or a multiple of 11.

Let's identify the digits at odd and even places in 8A51464 from the right:

• Odd places (1st, 3rd, 5th, 7th): 4 (1st), 4 (3rd), 1 (5th), A (7th)
• Even places (2nd, 4th, 6th, 8th): 6 (2nd), 4 (4th), 5 (6th), 8 (8th)

Sum of digits at odd places = $4 + 4 + 1 + A = 9 + A$

Sum of digits at even places = $6 + 4 + 5 + 8 = 23$

The difference is $(9 + A) - 23 = A - 14$.

For the number to be divisible by 11, this difference ($A - 14$) must be 0 or a multiple of 11.

Since A is a single digit (0 to 9), the value of $A - 14$ will be between $0 - 14 = -14$ and $9 - 14 = -5$.

The only multiple of 11 in the range [-14, -5] is -11.

So, we must have $A - 14 = -11$.

Solving for A:

$A = -11 + 14$

$A = 3$

So, A = 3.

Calculating the Value of AB

We have found that A = 3 and B = 4.

The question asks for the value of AB, which means $A \times B$.

Value of AB = $3 \times 4 = 12$.

Therefore, the value of AB is 12.

Revision Table: Key Divisibility Rules

Additional Information: Divisibility by Composite Numbers

When checking divisibility by a composite number (a number with more than two factors), you can often break it down into its co-prime factors. If a number is divisible by each of its co-prime factors, then it is divisible by the composite number itself.

• For example, to check divisibility by 88, we check divisibility by its co-prime factors, 8 and 11.
• To check divisibility by 6, we check divisibility by its co-prime factors, 2 and 3.
• To check divisibility by 12, we check divisibility by its co-prime factors, 3 and 4 (not 2 and 6, as they are not co-prime).

This principle was applied in solving the problem involving divisibility by 88.