2024 AMC 10A Problems/Problem 11

Problem

How many ordered pairs of integers $(m, n)$ satisfy $\sqrt{n^2 - 49} = m$ ?

$\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}$ Infinitely many

Solution

Note that $m$ is a nonnegative integer.

We square, rearrange, and apply the difference of squares formula to the given equation: $(n+m)(n-m)=49.$ It is clear that $n+m\geq n-m,$ so $(n+m,n-m)=(49,1),(7,7),(-7,-7),(-1,-49).$ Each ordered pair $(n+m,n-m)$ gives one ordered pair $(m,n),$ so there are $\boxed{\textbf{(D)}~4}$ such ordered pairs $(m,n).$

Problem

How many ordered pairs of integers $(m, n)$ satisfy $\sqrt{n^2 - 49} = m$ ?

$\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}$ Infinitely many

Solution 2

Squaring both sides of the given equation gives $n^2-49=m^2\rightarrow n^2-m^2=49\rightarrow (n+m)(n-m)=49$ Splitting $49$ into its factors (keep in mind it doesn't ask for positive integers, so the factors can be double negative, too) gives six cases:

$(1\cdot49)$

$(7\cdot7)$

$(49\cdot1)$

$(-1\cdot -49)$

$(-7\cdot -7)$

$(-49\cdot -1)$ .

Note that the square root in the problem doesn't have $\pm$ with it. Therefore, if there are two solutions, $(n,m)$ and $(n,-m)$ , then these together are to be counted as one solution. The solutions expressed as $(n,m)$ are:

$(25,24)$

$(25,-24)$

$(7,0)$

$(-7,0)$

$(-25,24)$

$(-25,-24)$ .

$(25,24)$ and $(25,-24)$ are to be counted as one, same for $(-25,24)$ and $(-25,-24)$ . Therefore, the solution is $\boxed{\text{(D) }4}$ ~Tacos_are_yummy_1

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2024 AMC 10A Problems/Problem 11

Contents

Problem

Solution

Problem

Solution 2