In a right triangle with integer length sides, the hypotenuse has length 39 units. How many units is the length of the shorter leg?
Since they only ask for one length, you can assume that there's only one possible triangle.  Then, quickly note that $39 = 3\cdot 13$, and that 5 - 12 - 13 is a Pythagorean triple.  Therefore, the shorter leg has length $\boxed{15}$.
15
Level 5
Prealgebra
Problem: In a right triangle with integer length sides, the hypotenuse has length 39 units. How many units is the length of the shorter leg?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the length of the shorter leg in a right triangle with a hypotenuse of 39 units, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). This can be expressed as: 

c^2 = a^2 + b^2.

Here, c = 39, so we calculate:

39^2 = a^2 + b^2.

Calculating 39^2 gives us:

1521 = a^2 + b^2.

We need to find integer values for a and b, where a is the shorter leg. Since a must be less than b, we can express b in terms of a:

b = sqrt(1521 - a^2).

To ensure b is an integer, 1521 - a^2 must be a perfect square. We can iterate through possible integer values for a, starting from 1 up to 39, and check if 1521 - a^2 results in a perfect square.

Testing values:

For a = 1: 1521 - 1^2 = 1520 (not a perfect square).
For a = 2: 1521 - 2^2 = 1517 (not a perfect square).
For a = 3: 1521 - 3^2 = 1512 (not a perfect square).
...
Continuing this process, we find:

For a = 15: 1521 - 15^2 = 1521 - 225 = 1296, which is 36^2 (a perfect square).

Thus, we have a = 15 and b = 36. Since we are looking for the length of the shorter leg, we conclude that the length of the shorter leg is 15 units.<|END_OF_LATENT|> The final answer is $15$.