Posted by **Lauren** on Saturday, February 26, 2011 at 1:55pm.

x is a number between 3 and 50.

How many vales of x is radical x / 13 a rational number?

- Math -
**MathMate**, Saturday, February 26, 2011 at 2:57pm
"x is a number between 3 and 50." requires any number x (integer, fractional or decimal) between the given limits such that √(x/13) is rational.

The number of integers is limited to 1, namely 13, since √(13/13)=1 is rational. Multiplying 13 by the nearest perfect square exceeds the upper limit.

However, there are many non-integer rational numbers that fit the bill.

Basically, we multiply 13 by fractions whose numerator and denominator are perfect squares.

13/4

13*4/9

13*9/4

13*25/9

13*9/25

13*25/36

13*36/25

13*36/49

13*49/36

13*49/64

13*64/49

...

As you can see, the list can go on and on, and the list is long, if not infinite.

You may want to double check if the question actually meant:

"x is an integer between 3 and 50."

- Math -
**Reiny**, Saturday, February 26, 2011 at 2:59pm
for √x/13 to be rational, x must be a perfect square.

perfect squares between 3 and 50 are

4,9,16,25,36, and 49

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