2020 AMC 10B Problems/Problem 24

Problem

How many positive integers $n$ satisfy $\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$ (Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$ .)

$\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$

Solution

(Quick solution if you’re in a hurry)

First notice that (n+1000)/70 must be an integer. This means that n is congruent to 50 (mod 70). Testing the first few values of n (adding 70 to n each time and noticing the left side increases by 1 each time) yields n=20 and n=21. At this point, the right side increases faster than the left.

However, since the left side is linear and the right side is roughly a radical, the graphs must intersect in at least one other point. Using binary search can narrow down the other cases, being n=47, n=48, n=49, and n=50. This is 6 cases, for an answer of C.

~DrJoyo

2020 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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2020 AMC 10B Problems/Problem 24

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