Question

A certain number written in a certain base is 144. Which of the following is always true?

• A. I. Square root of the number written in the same base is 12
• B. II. If base is increased by 2, the number becomes 100
• C. Neither I nor II
• D. Both I and II

Hint:

Convert base expressions to decimal and observe algebraic identities like perfect squares.

Answer 1

The Correct Option is D

“144” in base \( b \) means: \[ 1 \cdot b^2 + 4 \cdot b + 4 = b^2 + 4b + 4 = (b + 2)^2 \Rightarrow \text{Perfect square in every base} \] So statement I is always true.
Now increase base to \( b+2 \), then 144 becomes: \[ 1 \cdot (b+2)^2 + 4 \cdot (b+2) + 4 = (b+2)^2 + 4(b+2) + 4 = b^2 + 4b + 4 + 4b + 8 + 4 = b^2 + 8b + 16 = (b + 4)^2 \] Now if \( (b+4)^2 = 100 \Rightarrow b + 4 = 10 \Rightarrow b = 6 \), valid. So true for some base → satisfies condition.
Hence, both are valid: \[ \boxed{\text{Both I and II are true}} \]