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Number Theory (MA3Z7)
Problem Sheet V
1. Show that 
[Hint: d|n if and only if d/n|n.]
Deduce that 
2. Show that ϕ(n) is even for n ≥ 3.
3. Show that ϕ(d)|ϕ(n) whenever d|n.
4. A number n ∈ N is perfect if σ(n) = 2n, where σ(n) is the sum of the divisors of n. (eg. 1 + 2 + 3 + 6 = 2 × 6 so 6 is perfect.)
Show that if 2 p − 1 is prime, then 2 p−1 (2p − 1) is perfect.
[Hint: use the fact that σ(n) is multiplicative.]
More challenging: Show that every even perfect number is of this form.
Even more challenging: Does an odd perfect number exist?
5. Writing
(where the inner sum is over all divisors of (m, n)), deduce directly by swapping the order of summation that
