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A
 1, 2, 3, 4, 5, 6, 7, 8, 9
B
 1, 2, 3, 4, 5, 6, 7, 8, 9
C
 1, 2, 3, 4, 5, 6, 7, 8, 9
D
 1, 2, 3, 4, 5, 6, 7, 8, 9

 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
E
 1, 2, 3, 4, 5, 6, 7, 8, 9
F
 1, 2, 3, 4, 5, 6, 7, 8, 9

 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
G
 1, 2, 3, 4, 5, 6, 7, 8, 9

 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
H
 1, 2, 3, 4, 5, 6, 7, 8, 9

 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
I
 1, 2, 3, 4, 5, 6, 7, 8, 9

 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
J
 1, 2, 3, 4, 5, 6, 7, 8, 9

 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

5x5, not too difficult. (July 22)

No zero occurs in this puzzle.

Across:
A: A prime number
D: A across multiplied with the digit sum of D down
F: A prime number
G: A square number
H: A multiple of D across
J: A palindrome (a number that equals its reverse)
Down:
A: A multiple of D down
B: If you are a purist, ignore this piece of information since you don't need it: B down is prime.
C: The reverse of C down is a multiple of the digit sum of F across.
D: The reverse of A across
E: If you are a purist, ignore this piece of information since you don't need it: all digits of E down are even.
I: A square number