Question

The least number of five digit which is exactly divisible by 12, 15, and 18 is

• A. 10080
• B. 10015
• C. 10010
• D. 10020

Hint:

You can also solve this quickly by testing the options starting from the smallest value:
- $10010$ is not divisible by $15$ (does not end in $0$ or $5$).
- $10015$ is not divisible by $12$ or $18$ (not an even number).
- $10020$ is not divisible by $18$ (sum of digits $1+0+0+2+0 = 3$, not divisible by $9$).
- $10080$ ends in $0$ (divisible by $15$) and is even (divisible by $12$, $18$), and sum of digits is $9$ (divisible by $18$).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To find the smallest number of a certain digit length that is exactly divisible by a set of numbers, we first calculate the Least Common Multiple (LCM) of those numbers.
The LCM represents the smallest positive integer that is divisible by all numbers in the set.
Any other common multiple of these numbers must be a multiple of their LCM.
Key Formula or Approach:
1. Find the LCM of $12, 15,$ and $18$ using prime factorization.
2. Find the smallest 5-digit number, which is $10000$.
3. Divide $10000$ by the LCM to find the next higher multiple.

Step 2: Detailed Explanation:
Let us perform the calculations step-by-step:
1. Calculate the LCM of 12, 15, and 18:
- Write the prime factorizations: \[ 12 = 2^2 \times 3^1 \] \[ 15 = 3^1 \times 5^1 \] \[ 18 = 2^1 \times 3^2 \] - To find the LCM, take the highest power of each prime factor present: \[ \text{LCM} = 2^2 \times 3^2 \times 5^1 \] \[ \text{LCM} = 4 \times 9 \times 5 = 180 \] 2. Find the smallest 5-digit multiple of 180:
- The smallest 5-digit number is $10000$.
- Divide $10000$ by $180$: \[ \frac{10000}{180} \approx 55.55 \] - Since we need an integer multiple that is at least $10000$, we round up $55.55$ to the next integer, which is $56$.
- Multiply $180$ by $56$ to find the smallest 5-digit multiple: \[ 180 \times 56 = 10080 \] Therefore, $10080$ is the least 5-digit number exactly divisible by $12, 15,$ and $18$.

Step 3: Final Answer:
The least 5-digit number exactly divisible by 12, 15, and 18 is 10080.
Therefore, the correct option is (A).