Value of \(\mu(20)+\sigma(20)\) is (where \(\mu(n)\) is the Mobius function and \(\sigma(n)\) is the sum of all positive divisors of n)
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 42
The Mobius function is defined as:
\(\mu(1) = 1\)
\(\mu(n) = 0\) if \(p^2\) divides n for some prime p
\(\mu(n) = (-1)^k\) if n is a product of k distinct primes
\(20 = 2^2\times 5\)
The square \(2^2\) divides 20, so \(\mu(20) = 0\).
The positive divisors of 20 are 1, 2, 4, 5, 10, 20:
\(\sigma(20) = 1+2+4+5+10+20 = 42\)
\(\mu(20)+\sigma(20) = 0+42 = 42\)