The least number of five digit is exactly divisible by 88 is

## Concept & Logic This problem is based on the concept of finding a number within a specific range that is completely divisible by a given divisor. To find the least $n$-digit number divisible by a divisor $d$: 1. Identify the smallest $n$-digit number. 2. Divide it by $d$ to find the remainder. 3. Add the difference of the divisor and the remainder $(d - r)$ to the smallest $n$-digit number. ## Step-by-Step Solution * **Given:** We need the least 5-digit number divisible by $88$. * The smallest 5-digit number is $10000$. * **Calculation:** Divide $10000$ by $88$. $$10000 \div 88$$ $88 \times 113 = 9944$ * The remainder ($r$) is: $$10000 - 9944 = 56$$ * **Deduction:** If we subtract $56$ from $10000$, the number becomes a 4-digit number ($9944$), which is divisible by $88$. * To get the next multiple of $88$ (which will be the smallest 5-digit multiple), we must add the divisor $88$ to $9944$: $$9944 + 88 = 10032$$ * Alternatively, use the standard formula: Least Number + (Divisor - Remainder) $$10000 + (88 - 56) = 10000 + 32 = 10032$$ ## Exam Strategy & Shortcut Use the **Divisibility Rule Option Elimination** method. A number is divisible by $88$ if it is divisible by both $8$ and $11$. * **Rule of 8:** The last 3 digits must be divisible by 8. * (a) $032 \div 8 = 4$ (Divisible) * (b) $132 \div 8 = 16.5$ (Not divisible) * (c) $088 \div 8 = 11$ (Divisible) * (d) $023$ (Not even, so not divisible) * We are left with (a) $10032$ and (c) $10088$. * The question asks for the *least* number. $10032$ is smaller than $10088$. * Verify $10032$ with the **Rule of 11** (Difference of alternating sums is 0 or 11): $(1+0+2) - (0+3) = 3 - 3 = 0$. It works! ## Common Pitfall A very common mistake is simply subtracting the remainder ($56$) from $10000$ and arriving at $9944$, then forgetting that $9944$ is a 4-digit number. Always remember to add the divisor back or use the $(d - r)$ addition rule. ## Final Answer Therefore, the correct answer is 10032.