Problem
2025 AIME I Problem 10
The 27 cells of a 3 \times 9 grid are filled in using the numbers 1 through 9 so that each row contains 9 different numbers, and each of the three 3 \times 3 blocks heavily outlined in the example below contains 9 different numbers, as in the first three rows of a Sudoku puzzle.
![[asy] unitsize(20); add(grid(9,3)); draw((0,0)--(9,0)--(9,3)--(0,3)--cycle, linewidth(2)); draw((3,0)--(3,3), linewidth(2)); draw((6,0)--(6,3), linewidth(2)); real a = 0.5; label("5",(a,a)); label("6",(1+a,a)); label("1",(2+a,a)); label("8",(3+a,a)); label("4",(4+a,a)); label("7",(5+a,a)); label("9",(6+a,a)); label("2",(7+a,a)); label("3",(8+a,a)); label("3",(a,1+a)); label("7",(1+a,1+a)); label("9",(2+a,1+a)); label("5",(3+a,1+a)); label("2",(4+a,1+a)); label("1",(5+a,1+a)); label("6",(6+a,1+a)); label("8",(7+a,1+a)); label("4",(8+a,1+a)); label("4",(a,2+a)); label("2",(1+a,2+a)); label("8",(2+a,2+a)); label("9",(3+a,2+a)); label("6",(4+a,2+a)); label("3",(5+a,2+a)); label("1",(6+a,2+a)); label("7",(7+a,2+a)); label("5",(8+a,2+a)); [/asy]](/static/aops/df3daade257e4001f968e9c379a01ae9c94e8883.png)
The number of different ways to fill such a grid can be written as p^a \cdot q^b \cdot r^c \cdot s^d where p, q, r, and s are distinct prime numbers and a, b, c, d are positive integers. Find p \cdot a + q \cdot b + r \cdot c + s \cdot d.
Leading zeroes must be inputted, so if your answer is 34, then input 034
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