Consider a matrix nXn representing a multiplication table i.e. a_ij=i*j .
Determine the number of appearances of n in this matrix.

Assume n is prime, then it will appear twice (e,g. 1*7=7,7*1=7)

Assume n is compound with 2 distinct prime factors. Then it will appear twice (e,g. 1*6=6,6*1=6) and twice more at the junction of the factors (6=2*3, 6=3*2). With 3 distinct factors: twice, as usual, plus twice more for each combination of factors (30=2*15,30=15*2, 30=5*6,30=6*5, 30=3*10,30=10*3).

If the factors are the same, i.e. the number is a square: twice, as usual, plus once more for the square, since the squares fall on the main diagonal: (9=3*3). So the number of appearances of n depends on the factors of n.

Posted by broll on 2012-12-11 10:50:48