How many three-digit numbers can be expressed as an integral power of 2?

Correct Answer: C

Explanation: An integral power of 2 is any number that can be expressed in the format 2ⁿ, where n is an integer. To find how many such numbers are exactly three digits long (i.e., integers ranging from 100 to 999), we can systematically evaluate the positive powers of 2:

• 2¹ = 2
• 2² = 4
• 2³ = 8
• 2⁴ = 16
• 2⁵ = 32
• 2⁶ = 64
• 2⁷ = 128 (First three-digit power)
• 2⁸ = 256 (Second three-digit power)
• 2⁹ = 512 (Third three-digit power)
• 2¹⁰ = 1024 (First four-digit power)

As observed, only 128, 256, and 512 fall into the three-digit category. Therefore, exactly 3 three-digit numbers can be expressed as an integral power of 2.

Why the other options are incorrect:

• Option A (1): Incorrect because it assumes there is only one valid number, ignoring the other two possible powers within the 100–999 range.
• Option B (2): Incorrect as it counts only two numbers, likely omitting either the lowest (2⁷ = 128) or highest (2⁹ = 512) valid power.
• Option D (4): Incorrect because counting four numbers would require including 2¹⁰ = 1024, which is a four-digit number.

Takeaway: Memorising the fundamental powers of 2 up to 2¹⁰ (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024) is highly advantageous for Number System and logical reasoning questions in the UPSC CSAT. It saves calculation time and helps you rapidly identify mathematical patterns.