- A17
- B19
- C36
- D18
Step-by-Step Solution
Between and there are exactly natural numbers. Here , so .
Answer: D) 18
How many natural numbers lie between the squares and ?
Class 8Squares and Square RootsSquares and Roots
Key idea: Properties of perfect squares
Useful patterns to spot a perfect square (and reject non-squares):
- **Last digit**: a perfect square ends in 0, 1, 4, 5, 6, or 9 — NEVER in 2, 3, 7, or 8. So 142 cannot be a perfect square (ends in 2).
- **Number of zeros at the end**: must be even. 100 is a perfect square ; 1000 is not.
- **Squares of even numbers are even; squares of odd numbers are odd.**
- **Sum of consecutive odd numbers**: 1 + 3 + 5 + … + . Try it: .
These let you reject many candidates without doing any heavy work.
- **Last digit**: a perfect square ends in 0, 1, 4, 5, 6, or 9 — NEVER in 2, 3, 7, or 8. So 142 cannot be a perfect square (ends in 2).
- **Number of zeros at the end**: must be even. 100 is a perfect square ; 1000 is not.
- **Squares of even numbers are even; squares of odd numbers are odd.**
- **Sum of consecutive odd numbers**: 1 + 3 + 5 + … + . Try it: .
These let you reject many candidates without doing any heavy work.
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