Question

Let $(a,b,c,d)$ be an ordered quadruple of not necessarily distinct integers, each one of them in the set $\{0,1,2,3\}.$ For how many such quadruples is it true that $a\cdot d-b\cdot c$ is odd? (For example, $(0,3,1,1)$ is one such quadruple, because $0\cdot 1-3\cdot 1 = -3$ is odd.)

$48$

$64$

$96$

$128$

$192$

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Written solution:

Only parity matters. Modulo $2$ , the condition is that $ad-bc$ is $1$ , meaning the matrix $\begin{pmatrix}a&b\\ c&d\end{pmatrix}$ is invertible over $\mathbb F_2$ .

There are $(4-1)(4-2)=6$ invertible $2\times2$ matrices over $\mathbb F_2$ . Each parity pattern lifts to $2^4=16$ choices from $\{0,1,2,3\}$ , so there are $6\cdot16=96$ quadruples. Thus, C is the correct answer.

2020 AMC 10A Problem 18

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Problem 18 in Other Years