Question

Calculate d(42), sigma(42), d(420), and sigma(420). Calculate d(540), sigma(540), d(5400), and sigma(5400). Calculate d and sigma of 10115 = 5 middot 7 middot 17^2 and 100115 = 5 middot 20023. Calculate d and sigma of 10116 = 2^2 middot 3^2 middot 281 and 100116 = 2s middot 3^5 Show that sigma(n) is odd if n is a power of two. Prove that f(n) is multiplicative, then so is f(n)/n. What is the smallest integer n such that d(n) = 8? Such that d(n) Does d(n) = k have a solution n for each k? In 1644, Mersenne asked for a number with 60 divisors. Find one than 10,000. Find infinitely many n such that d(n) = 60.

Answer 1

To calculate the number of divisors (d) and the sum of divisors (sigma) for given numbers, we count the divisors and add them up. For example, d(42) is 3, sigma(42) is 45. For d(420) it is 18, and sigma(420) is 1056.

Step-by-step explanation:

To calculate d(42), we count the number of positive divisors, which is 3. The divisors are 1, 2, and 42. The sum of divisors, sigma(42), is 45.

To calculate d(420), we count the number of positive divisors, which is 18. The divisors are 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, and 70. The sum of divisors, sigma(420), is 1056.

To calculate d(540), we count the number of positive divisors, which is 32. The divisors are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, 270, 540. The sum of divisors, sigma(540), is 1491.

Similarly, to calculate d(5400), we count the number of positive divisors, which is 72. The sum of divisors, sigma(5400), is 23976.