4. Every integer N > 1 can be written as N = p1^a1 p2^a2 p3^a3 ... pk^ak, where k is a positive integer, p1 < p2 < p3 < ... < pk are prime numbers, and a1, a2, a3, ..., ak are positive integers. For example, 1400 = 2^3 5^2 7^1. The number of positive divisors of N is denoted by f(N). It is known that f(N) = (1 + a1)(1 + a2)(1 + a3) ... (1 + ak) (a) How many positive divisors does 240 have? That is, what is the value of f(240)? (b) Define an integer N > 1 to be refactorable if it is divisible by f(N). For example, both 6 and 8 have 4 positive divisors, so 8 is refactorable and 6 is not refactorable. This is because 8 is divisible by 4, but 6 is not divisible by 4. Determine all refactorable numbers N with f(N) = 6. (c) Determine the smallest refactorable number N with f(N) = 256. (d) Show that for every integer m > 1, there exists a refactorable number N such that f(N) = m.
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a) 240 has 4 positive divisors: 235, 271, 288, and 312. Show moreā¦
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