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1999 CEMC Pascal Problems/Problem 21

Problem

A number is Beprisque if it is the only natural number between a prime number and a perfect square (ex. $10$ is Beprisque but $12$ is not). The number of two-digit Beprisque numbers (including $10$) is

$\text{ (A) }\ 1 \qquad\text{ (B) }\ 2 \qquad\text{ (C) }\ 3 \qquad\text{ (D) }\ 4 \qquad\text{ (E) }\ 5$

Solution 1

We can first list all of the perfect squares from $9$ to $100$, and then list of all the prime numbers between them:

$9, 16, 25, 36, 49, 64, 81, 100$

The prime numbers between $9$ and $16$ are $11$ and $13$.

The prime numbers between $16$ and $25$ are $17$, $19$, and $23$.

The prime numbers between $25$ and $36$ are $29$ and $31$.

The prime numbers between $36$ and $49$ are $37$, $41$, $43$, and $47$

The prime numbers between $49$ and $64$ are $53$, $59$, and $61$.

The prime numbers between $64$ and $81$ are $67$, $71$, $73$, and $79$.

The prime numbers between $81$ and $100$ are $83$, $89$, and $97$.

We can notice that $10$, $24$, $48$, $80$, and $82$ all fit the criteria. The number of numbers listed is $\boxed {\textbf {(E) } 5}$.

~anabel.disher

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