Question

Find the smallest positive integer k such that 3240/k is a square number

Answer 1

Answer:

10

Step-by-step explanation:

First, factorize the number 3,240:

$3,240=2\cdot 1,620=2\cdot 2\cdot 810=2\cdot 2\cdot 2\cdot 405=2\cdot 2\cdot 2\cdot 2\cdot 5\cdot 81=2^3\cdot 5\cdot 9^2$

So,

$3,240=2^2 \cdot 2\cdot 5\cdot 9^2=18^2\cdot 10$

Now consider the fraction

$\dfrac{3,240}{k}=\dfrac{18^2\cdot 10}{k}$

If k = 10, then

$\dfrac{3,240}{k}=\dfrac{18^2\cdot 10}{10}=18^2$

is a square number.

For all k < 10, the fraction tex]\dfrac{3,240}{k}[/tex] is not a square number (this follows from factorization).

Or you can simply check the values of the fraction for all k < 10:

• k = 1, $\dfrac{3,240}{1}=3,240$ is not a square number;
• k = 2, $\dfrac{3,240}{2}=1,620$ is not a square number;
• k = 3, $\dfrac{3,240}{3}=1,080$ is not a square number;
• k = 4, $\dfrac{3,240}{4}=810$ is not a square number;
• k = 5, $\dfrac{3,240}{5}=648$ is not a square number;
• k = 6, $\dfrac{3,240}{6}=540$ is not a square number;
• k = 7, $\dfrac{3,240}{7}$ is not a square number;
• k = 8, $\dfrac{3,240}{8}=405$ is not a square number;
• k = 9, $\dfrac{3,240}{9}=360$ is not a square number.