Divisors
M117, September 21, 2011
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The number of divisors of a positive integer n is the number of positive integers a such that
a|n. (Note that a does NOT have to be prime to be a divisor.) For example, the number
6 has 4 divisors: 1, 2, 3, and 6. Answer the following questions about divisors of positive
integers. (Justify your answers.)
(1) How many divisors does the number 10 have?
(2) How many divisors does the number 24 have?
(3) What kinds of numbers have exactly 2 divisors?
(4) What kinds of numbers have exactly 3 divisors?
(5) What kinds of numbers have exactly 4 divisors?
(6) Find the smallest positive integer with exactly 5 divisors.
(7) Find the smallest positive integer with exactly 6 divisors.
(8) Let n have prime factorization pa q b . How many divisors does n have?
(9) Find the smallest positive integer with exactly 12 divisors.