- A19
- B21
- C20
- D100
Step-by-Step Solution
The th layer has the th odd number of squares, . So the 10th layer has squares (and the total becomes ).
Answer: A) 19
The picture builds a square from L-shaped layers of 1, 3, 5 and 7 small squares. If the pattern continues, how many small squares will the 10th layer have?
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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