Convert 2710 to another number in base three

Analyze the question: "2710" is most likely 27 in base ten (27₁₀), and we want it in base three. The options carry a subscript 3, such as 1000₃.

Concept: To convert a base-10 number to base 3, divide repeatedly by 3 and record the remainders. Read the remainders from bottom to top.

Solution:

• 27 ÷ 3 = 9 remainder 0
• 9 ÷ 3 = 3 remainder 0
• 3 ÷ 3 = 1 remainder 0
• 1 ÷ 3 = 0 remainder 1

Reading from bottom to top: 1000₃

Check: 1000₃ = 1×3³ + 0×3² + 0×3¹ + 0×3⁰ = 27 ✓

Compare with the other options:

• 1001₃ = 27 + 1 = 28
• 1010₃ = 27 + 3 = 30
• 1100₃ = 27 + 9 = 36

None of these equals 27, so only 1000₃ works.

Answer: D.

Tip: Powers of the base are always written as 1 followed by zeros in that base. Since 27 = 3³, it is 1000 in base three. Likewise, 8 = 2³ is 1000 in base two. Learn the powers of 3 (1, 3, 9, 27, 81) and you can often convert without long division.