2006 iTest Problems/Problem 15

Problem

How many integers between $1$ and $2006$, inclusive, are perfect squares?

$\text{(A) }37\qquad \text{(B) }38\qquad \text{(C) }39\qquad \text{(D) }40\qquad \text{(E) }41\qquad \text{(F) }42\qquad \text{(G) }43\qquad \text{(H) }44\qquad$

$\text{(I) }45\qquad \text{(J) }46\qquad \text{(K) }47\qquad \text{(L) }48\qquad \text{(M) }49\qquad \text{(N) }50\qquad \text{(O) }\text{none of the above}\qquad$

Solution

Note that $1600 < 2006 < 2500$, so the number of perfect squares must be around 40 to 50. With some trial and error, we found that $44^2 = 1936$ and $45^2 = 2025$, so the perfect squares between 1 and 2006 are $1^2, 2^2, 3^2, \cdots 44^2$. There are $\boxed{\textbf{(H) } 44}$ integers in the set.

See Also

2006 iTest (Problems)
Preceded by:
Problem 14
Followed by:
Problem 16
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