If you multiply a blue number by an orange number, what color will the product be?

Assume that all integers are colored according to the same pattern as the numbers in the grid.

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The blue numbers are the odd numbers, and the orange numbers are the even numbers, so the question can be rephrased as follows: If you multiply an odd number by an even number, will the product be odd or even?

The product of an odd number and an even number is always even, regardless of the specific values of the numbers. An odd number must be of the form $\fn_phv \LARGE {\color{Blue} 2j+1}$ and an even number must be of the form $\fn_phv \LARGE {\color{Blue} 2k}$ (where $\fn_phv \LARGE {\color{Blue} j}$ and  $\fn_phv \LARGE {\color{Blue} k}$are integers); the product of these numbers is

$\fn_phv \LARGE {\color{Blue} (2j+1)\times 2k=(2j\times 2k)+2k=4jk+2k=2(2jk+k)}$

Since $\fn_phv \LARGE {\color{Blue} 2}$ divides this product, it must be even. This reasoning does not depend on the specific values of $\fn_phv \LARGE {\color{Blue} j}$ and $\fn_phv \LARGE {\color{Blue} k}$ so this result is true for all $\fn_phv \LARGE {\color{Blue} j}$ and $\fn_phv \LARGE {\color{Blue} k}$ thus, the product of an odd number and an even number is always even, and therefore, if you multiply a blue number by an orange number, the product will be orange.

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