20. Each square in this cross-number can be filled with a non-zero digit such that all of the conditions in the clues are fulfilled. The digits used are not necessarily distinct.
What is the answer to 3 ACROSS?

ACROSS
1. A composite factor of 1001
3. Not a palindrome
5. $p q^{3}$ where $p, q$ prime and $p \neq q$
DOWN
1. One more than a prime, one less than a prime
2. A multiple of 9
4. $p^{3} q$ using the same $p, q$ as 5 ACROSS
Solution
295
1 Across may be either 77 or 91 . The only possibility for 1 Down with 7 or 9 as its first digit is 72. So 1 Across is 77 and 1 Down is 72 .

In the clues for 5 Across and 4 Down we see that $p, q$ must be 2,3 in some order, since if any larger prime were used then $p q^{3}$ and $q p^{3}$ would not both be two-digit. Therefore, 5 Across and 4 Down are $3 \times 2^{3}=24$ and $2 \times 3^{3}=54$ in some order. We know that 3 Across is not a palindrome (so may not end in a 2). Therefore, 5 Across is 24 and 4 Down is 54.
The only three-digit multiples of 9 beginning with a 7 are 702 and 792. As every digit in the completed crossnumber must be non-zero we have 2 Down is 792 and 3 Across is 295 .