Select the pair of numbers that does NOT exhibit the same mathematical relationship as the rest.

The correct option is 13, 156.

To determine which pair does not share the same mathematical relationship, let us analyze the pattern followed by each pair of numbers:

For the pair (8, 64):

$8^2=64$

The second number is the square of the first number.

For the pair (5, 25):

$5^2=25$

The second number is the square of the first number.

For the pair (12, 144):

${12}^2=144$

The second number is the square of the first number.

Now, let us examine the pair (13, 156):

${13}^2=169$

The square of 13 is 169. However, the second number provided in this pair is 156, which is $13\times 12=156$ instead of 169.

Therefore, the pair 13, 156 does NOT exhibit the square relationship ($x\to x^2$) shared by the rest of the options.