If 9A4157B is divisible by 88, then what is the unit digit of A x B?Options :08415

To determine the unit digit of $A \times B$ in the number $9A4157B$ when it is divisible by $88$, we must check its divisibility by both $8$ and $11$ since $88 = 8 \times 11$.

1. Break Down the Problem

1. Check divisibility by $8$.
2. Check divisibility by $11$.
3. Find values for digits $A$ and $B$ that satisfy both conditions.
4. Calculate $A \times B$ and find its unit digit.

2. Relevant Concepts

• Divisibility by 8: A number is divisible by 8 if its last three digits form a number that is divisible by 8.
• Divisibility by 11: A number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is either $0$ or a multiple of $11$.

3. Analyze Each Part

Divisibility by 8: The last three digits of the number are $57B$. We need to find $B$ such that $57B$ is divisible by $8$:

• $570 \div 8 = 71.25$ (not divisible)
• $571 \div 8 = 71.375$ (not divisible)
• $572 \div 8 = 71.5$ (not divisible)
• $573 \div 8 = 71.625$ (not divisible)
• $574 \div 8 = 71.75$ (not divisible)
• $575 \div 8 = 71.875$ (not divisible)
• $576 \div 8 = 72$ (divisible)

Thus, $B = 6$ is the only value making the last three digits $576$ divisible by $8$.

Divisibility by 11: Now the number is $9A41576$. We apply the divisibility rule:

Odd position digits: $9 + 4 + 7 = 20$
Even position digits: $A + 1 + 6 = A + 7$

Calculate the difference: $20 - (A + 7) = 13 - A$ To be divisible by $11$: $13 - A \equiv 0 \mod 11$ Thus, $A = 2 \text{ or } 13$ Since $A$ must be a single-digit number, $A = 2$.

4. Verify and Summarize

We found $A = 2$ and $B = 6$. Now we compute $A \times B$: $A \times B = 2 \times 6 = 12$ The unit digit of $12$ is $2$.

Final Answer

The unit digit of $A \times B$ is $2$.