2015 · Q36
Consider the following statements for the sequence of numbers given below : 11, 111, 1111, 11111,.....
- 1Each number can be expressed in the form (4m+3), where m is a natural number.
- 2Some numbers are squares.
Which of the above statements is/are correct?
Elementary MathematicsNumber System
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Explanation
Each repunit is ≡ 3 (mod 4): 11 ≡ 3, 111 ≡ 3, and for longer ones 10ᵏ ≡ 0 (mod 4) for k ≥ 2, so every term ≡ 1 + 2 = 3 = 4m+3 — statement 1 is true. Squares are 0 or 1 (mod 4), never 3 — statement 2 is false. Hence '1 only' (option A).