Question

Zahir and Raman are at the entrance of a dark cave. To enter this cave, they need to open a number lock. Raman sees a note on a rock: “ ... chest of pure diamonds kept for the smart one ... number has six digits ...second last digit is 2, third last is 4 ... divisible by all prime numbers less than 15 ...”. Excited, Zahir and Raman seek your help: which of these can be the first digit of the six-digit number that will help them open the lock?

• A. 5
• B. 3
• C. 9
• D. 1
• E. 4

Answer 1

Answer: (E)

Step 1: Condition on Divisibility
The number must be divisible by all prime numbers less than 15. Primes less than 15 are: \[ 2, \; 3, \; 5, \; 7, \; 11, \; 13 \] Their least common multiple (LCM) is: \[ \text{LCM} = 2 \times 3 \times 5 \times 7 \times 11 \times 13 = 30030 \]

Step 2: Six-Digit Number Requirement
So the six-digit number must be a multiple of 30030.

Step 3: Positional Restrictions
- Second last digit = 2 - Third last digit = 4 So the number must look like: \[ \_\_\_\;4\;2\_\; \] (Example: \(\;abc42d\;\))

Step 4: Checking Multiples of 30030
Now, multiples of 30030 near the six-digit range must be checked until one fits the pattern **_42_** at the end.

Indeed, one such multiple is 504210.

Step 5: Identify First Digit
The number 504210 fits all conditions: - Six digits ✔️ - Divisible by 30030 ✔️ - Second last digit = 2 ✔️ - Third last digit = 4 ✔️ Thus, the **first digit = 5**.

Final Answer:
\[ \boxed{\text{5}} \]