Let p and p + 2 be prime numbers and let Δ=| ccc p ! (p+1) ! (p+2) ! (p+1) ! (p+2) ! (p+3) ! (p+2) ! (p+3) ! (p+4) ! | Then the sum of the maximum values of α and β, such that p^(α) and (p+2)^(β) divide Δ, is _ _ _ _ _.

Matrices and Determinants: JEE Main 2022 Mathematics question

JEE Main 2022 · 29 Jul, Morning shift · Q37

JEE Main 202229 Jul, Morning shiftMathematicsHardMatrices and Determinants245× since 2002

Let p and p + 2 be prime numbers and let

\Delta=\left|\begin{array}{ccc} \mathrm{p} ! & (\mathrm{p}+1) ! & (\mathrm{p}+2) ! \\ (\mathrm{p}+1) ! & (\mathrm{p}+2) ! & (\mathrm{p}+3) ! \\ (\mathrm{p}+2) ! & (\mathrm{p}+3) ! & (\mathrm{p}+4) ! \end{array}\right|

Then the sum of the maximum values of

\alpha

and

\beta

, such that

\mathrm{p}^{\alpha}

and

(\mathrm{p}+2)^{\beta}

divide

\Delta

, is __________.

Your answerType your answer. Integers and decimals are accepted.