20. Each cell in this cross-number can be filled with a non-zero digit such that all of the conditions in the clues are satisfied. The digits used are not necessarily distinct. Determine T, the sum of all six answers to the clues. The answer to this question is $0.5 \times T$.

ACROSS:
1. Two less than a prime.
3. Exactly 100 more than its largest proper factor.
5. A multiple of 13 .

DOWN:
1: A fourth power.
2: A cube.
4: Not prime, not square, not even.
SolUTION
582
1 DOWN is either 16 or 81 . If 3 ACROSS begins with a six then its largest proper factor is at most 349 , which violates the clue. Therefore 3 ACROSS begins with a one, and 1 DOWN must be 81 .
1 ACROSS is either 81 or 87 . If 2 DOWN begins with a one then it is 125 and 5 ACROSS must be 52. But this would mean 4 DOWN ended with a two, which violates its clue. Therefore 1 ACROSS must be 87 , in turn meaning 2 DOWN is 729 and 5 ACROSS is 91 .
4 DOWN ends in a one so can only be 21,51 or 91 . This means 3 ACROSS has possible answers of 122,125 and 129 . Of these, only 125 satisfies the clue.
The total number of all the answers in the grid is $87+125+91+81+729+51=1164$. The required quantity is $1164 \div 2=582$.