Finding Numbers Divisible by 6 and 31

To determine which number is divisible by both 6 and 31, we must check each option against the divisibility rules for 6 and then test for divisibility by 31.

Understanding Divisibility Rules

• Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
  • Rule for 2: The number must be even (end in 0, 2, 4, 6, or 8).
  • Rule for 3: The sum of the number's digits must be divisible by 3.
• Divisibility by 31: There isn't a simple divisibility rule for 31. We check this by dividing the number by 31. If the division results in a whole number (an integer), the number is divisible by 31.

Analyzing Each Option

Option 1: 3720

• Check Divisibility by 6:
  • Is 3720 even? Yes, it ends in 0. So, it's divisible by 2.
  • Sum of digits = $3 + 7 + 2 + 0 = 12$.
  • Is 12 divisible by 3? Yes, $12 \div 3 = 4$.
  • Since 3720 is divisible by both 2 and 3, it is divisible by 6.
• Check Divisibility by 31:
  • Divide 3720 by 31: $3720 \div 31$.
  • Calculation: $3720 \div 31 = 120$.
  • The result (120) is a whole number, so 3720 is divisible by 31.
• Result for 3720: Divisible by both 6 and 31.

Option 2: 2132

• Check Divisibility by 6:
  • Is 2132 even? Yes, it ends in 2. So, it's divisible by 2.
  • Sum of digits = $2 + 1 + 3 + 2 = 8$.
  • Is 8 divisible by 3? No, $8 \div 3$ is not a whole number.
  • Since 2132 is not divisible by 3, it is not divisible by 6.
• Check Divisibility by 31:
  • Divide 2132 by 31: $2132 \div 31$.
  • Calculation: $2132 \div 31 \approx 68.77$.
  • The result is not a whole number, so 2132 is not divisible by 31.
• Result for 2132: Not divisible by 6 or 31.

Option 3: 4916

• Check Divisibility by 6:
  • Is 4916 even? Yes, it ends in 6. So, it's divisible by 2.
  • Sum of digits = $4 + 9 + 1 + 6 = 20$.
  • Is 20 divisible by 3? No, $20 \div 3$ is not a whole number.
  • Since 4916 is not divisible by 3, it is not divisible by 6.
• Check Divisibility by 31:
  • Divide 4916 by 31: $4916 \div 31$.
  • Calculation: $4916 \div 31 \approx 158.58$.
  • The result is not a whole number, so 4916 is not divisible by 31.
• Result for 4916: Not divisible by 6 or 31.

Option 4: 2858

• Check Divisibility by 6:
  • Is 2858 even? Yes, it ends in 8. So, it's divisible by 2.
  • Sum of digits = $2 + 8 + 5 + 8 = 23$.
  • Is 23 divisible by 3? No, $23 \div 3$ is not a whole number.
  • Since 2858 is not divisible by 3, it is not divisible by 6.
• Check Divisibility by 31:
  • Divide 2858 by 31: $2858 \div 31$.
  • Calculation: $2858 \div 31 \approx 92.19$.
  • The result is not a whole number, so 2858 is not divisible by 31.
• Result for 2858: Not divisible by 6 or 31.

Conclusion

After checking all the options using the divisibility rules, only the number 3720 meets both criteria: it is divisible by 6 and divisible by 31.