Compute ∅ (40), (124), (124) and check the equality in Σ∅() = 40.
Here's what I've done so far:
Not really sure about the summation equality.
∅ (40) = ∅ (5) x ∅ (8) => 4x4 = 16
(124) = 1+2+4+1+62+124 = 224
(124) = 1,2,4,31,62,124 = 6
Σ∅() = 40 => [Not sure how to compute this]
The summation equality says that for any natural number $n$, $$n = \varphi d + \varphi d' + \varphi d'' + \cdots$$ where $d, d', d''$ etc. are all the positive divisors of $n$. Since the divisors of $40$ are $1, 2, 4, 5, 8, 10, 20, 40$, you need to check that $$40 = \varphi 1 + \varphi 2 + \varphi 4 + \varphi 5 + \varphi 8 + \varphi 10 + \varphi 20 + \varphi 40$$