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a) Explain why the multiplicativity of p(n) follows from the formula p(n) = n* -II (1 - 1/p) where the product runs over the prime P factors of n. b) Prove this product formula for p(n) by induction on the number of prime-power factors of n, using as your base step the formula p(p^k) = p^k - p^(k-1) for primes p.

a) Explain why the multiplicativity of p(n) follows from the formula p(n) = n* -II (1 - 1/p) where the product runs over the prime P factors of n. b) Prove this product formula for p(n) by induction on the number of prime-power factors of n, using as your base step the formula p(p^k) = p^k - p^(k-1) for primes p.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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a) Explain why the multiplicativity of p(n) follows from the formula p (n) = n* II (1 - 1/p) where the product runs over the prime factors of n. b) Prove this product formula for p(n) by induction on the number of prime-power factors of n, using as your base step the formula p(p^k) = p^k - p^(k-1) for primes p.
Transcribed Image Text:a) Explain why the multiplicativity of p(n) follows from the formula p (n) = n* II (1 - 1/p) where the product runs over the prime factors of n. b) Prove this product formula for p(n) by induction on the number of prime-power factors of n, using as your base step the formula p(p^k) = p^k - p^(k-1) for primes p.
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