Which of the following numbers is completely divisible by 6?

Question

Which of the following numbers is completely divisible by 6?

1. A. 5,47,821
2. B. 7,53,216 (Correct answer)
3. C. 8,91,452
4. D. 6,42,538

Correct Answer

Option B — 7,53,216

Detailed Solution & Explanation

The correct answer is 7,53,216.

Shortcut Trick

• $6 = 2 \times 3$, so the number must be even and have a digit sum divisible by 3.
• Eliminate on the easier test first — parity: 5,47,821 ends in 1, so it fails immediately.
• Of the remaining even numbers, check digit sums:
  • $7{,}53{,}216 \Rightarrow 7+5+3+2+1+6 = 24$ ✓ divisible by 3
  • $8{,}91{,}452 \Rightarrow 29$ ✗
  • $6{,}42{,}538 \Rightarrow 28$ ✗
• Only 7,53,216 passes both.

Alternate Method

• Direct division confirms it: $753216 \div 6 = 125536$ exactly, with no remainder.

Additional Information

• Test the cheapest condition first. Checking parity takes a glance and removed one option here before any addition was needed.
• Composite divisors split into coprime factors: $6 = 2 \times 3$, $12 = 3 \times 4$, $15 = 3 \times 5$, $24 = 3 \times 8$. Splitting $12$ as $2 \times 6$ would be invalid, since 2 and 6 share a factor.
• A number divisible by 6 is automatically divisible by 2 and 3, but the converse needs both — being even alone is never sufficient.