BdMO Online Forum

Mehrab4226 wrote: ↑

Sat Apr 10, 2021 12:31 pm

How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$?

Pro_GRMR wrote: ↑

Sat Apr 10, 2021 12:50 pm

Mehrab4226 wrote: ↑

Sat Apr 10, 2021 12:31 pm

How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$?

According to the question:
$\frac{1}{2}mn = p \leq 80$ Such that $p$ is prime.
So, $\frac{1}{2}mn \in \{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79\} $
and $mn \in \{4, 6, 10, 14, 22, 26, \dots , 158\} $
$m | a$ Such that $a \in \{4, 6, 10, 14, 22, 26, \dots , 158\} $
There are $3$ divisors of $4$ and all the other numbers have $4$ divisors each. Because there are $22$ such numbers the result is $21 \times 4 + 3 = \boxed{87}$

Mehrab4226 wrote: ↑

Sat Apr 10, 2021 1:38 pm

Mehrab4226 wrote: ↑

Sat Apr 10, 2021 12:31 pm

How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$?

Aren't $(m,n)$ and $(n,m)$ the same triangle?

Pro_GRMR wrote: ↑

Sat Apr 10, 2021 5:19 pm

Mehrab4226 wrote: ↑

Sat Apr 10, 2021 1:38 pm

Mehrab4226 wrote: ↑

Sat Apr 10, 2021 12:31 pm

How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$?

Aren't $(m,n)$ and $(n,m)$ the same triangle?

Yeah but the question specifically asked for ordered pairs.
Let me give an example:
Let's say $\frac{1}{2}mn=3$. These four ordered pairs satisfy such constraint $(1, 6), (6, 1), (2, 3), (3, 2)$. Here the Order of the integers matter because they are ordered.