Divisibility Check for 6,30,605

The question asks us to find which number among the options – 3, 5, 7, or 14 – divides the number $6,30,605$ without leaving a remainder. We can use the divisibility rules to check this efficiently.

Applying Divisibility Rules

Let's test the number $6,30,605$ against each option using standard divisibility rules:

1. Divisibility by 3

• Rule: A number is divisible by 3 if the sum of its digits is a multiple of 3.
• Check: Sum the digits of $6,30,605$: $6 + 3 + 0 + 6 + 0 + 5 = 20$
• Conclusion: Since 20 is not divisible by 3, the number $6,30,605$ is not divisible by 3.

2. Divisibility by 5

• Rule: A number is divisible by 5 if its last digit is 0 or 5.
• Check: The last digit of $6,30,605$ is 5.
• Conclusion: As the last digit is 5, the number $6,30,605$ is divisible by 5. (Actual division: $6,30,605 \div 5 = 126,121$)

3. Divisibility by 7

• Rule: Take the last digit, double it, and subtract it from the number formed by the remaining digits. If the result is divisible by 7, the original number is divisible by 7. Repeat as necessary.
• Check: For $6,30,605$:
  • Remove the last digit (5) and double it: $2 \times 5 = 10$.
  • Subtract this from the remaining number ($63060$): $63060 - 10 = 63050$.
  • Repeat: Remove the last digit (0) and double it: $2 \times 0 = 0$.
  • Subtract from the remaining number ($6305$): $6305 - 0 = 6305$.
  • Repeat: Remove the last digit (5) and double it: $2 \times 5 = 10$.
  • Subtract from the remaining number ($630$): $630 - 10 = 620$.
  • Repeat: Remove the last digit (0) and double it: $2 \times 0 = 0$.
  • Subtract from the remaining number ($62$): $62 - 0 = 62$.
• Conclusion: Since 62 is not divisible by 7, the number $6,30,605$ is not divisible by 7.

4. Divisibility by 14

• Rule: A number is divisible by 14 if it's divisible by both 2 and 7.
• Check:
  • Divisibility by 2: A number is divisible by 2 if its last digit is even. The last digit of $6,30,605$ is 5 (odd). So, it is not divisible by 2.
  • Divisibility by 7: We already found $6,30,605$ is not divisible by 7.
• Conclusion: Since $6,30,605$ is not divisible by 2 (or 7), it cannot be divisible by 14.

Summary of Findings

By applying the divisibility rules:

• $6,30,605$ is NOT divisible by 3.
• $6,30,605$ IS divisible by 5.
• $6,30,605$ is NOT divisible by 7.
• $6,30,605$ is NOT divisible by 14.

Therefore, the number that divides $6,30,605$ among the given options is 5.