A perfect square has 5 digits. How many digits will its square root have?

Class 8Squares and Square RootsSquares and Roots

A2
B5
C2.5
D3

Step-by-Step Solution

Pairing a 5-digit number from the right gives three periods (the leftmost holding a single digit), and long division produces one quotient digit per period — so the root has 3 digits. Check the extremes:

10000 = 100

and

99856 = 316

, both 3-digit. The trap 2.5 halves 5; when the digit count is odd you must round UP, never take half a digit.

Answer: D) 3

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

(1 0^{2})

; 1000 is not.
- **Squares of even numbers are even; squares of odd numbers are odd.**
- **Sum of consecutive odd numbers**: 1 + 3 + 5 + … +

(2 n - 1) = n^{2}

. Try it:

1 + 3 + 5 = 9 = 3^{2}

.

These let you reject many candidates without doing any heavy work.

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