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Chapter 4: Problem 26

When does a positive integer \(n\) have exactly a) two positive divisors? b) three positive divisors? c) four positive divisors? d) five positive divisors?

Short Answer

Expert verified
a) For an integer to have exactly two positive divisors, it must be a prime number. b) To have exactly three positive divisors, the number must be the square of a prime number. c) To have exactly four divisors, the number must be a product of two distinct prime numbers or the cube of a prime number. d) For five divisors, the number must be the fourth power of a prime number.

Step by step solution

01

Finding when a positive integer \(n\) has exactly two positive divisors

A number has exactly two positive divisors - 1 and the number itself, only when it is a prime number. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Therefore, for a positive integer \(n\) to have exactly two positive divisors, it must be a prime number.
02

Finding when a positive integer \(n\) has exactly three positive divisors

For a positive integer \(n\) to have exactly three positive divisors, it must be the square of a prime number. In this case, the three divisors will be 1, the prime number and the square of the prime number.
03

Finding when a positive integer \(n\) has exactly four positive divisors

A number \(n\) has exactly four positive divisors in two cases: either it is the cube of a prime number or it is the product of two different prime numbers. In the first case, the divisors are 1, the prime number, the square of the prime number, and the cube of the prime number. In the second case, the divisors are 1, the two prime numbers, and their product.
04

Finding when a positive integer \(n\) has exactly five positive divisors

For a positive integer \(n\) to have exactly five positive divisors, it must be a number which is the fourth power of a prime number. In this case, the five divisors are 1, the prime number, the square of the prime number, the cube of the prime number, and the fourth power of the prime number.

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Most popular questions from this chapter

For all \(n \in \mathbf{Z}^{+}\), let \(S(n)\) be the open statement: \(n^{2}+n+41\) is prime. a) Verify that \(S(n)\) is true for all \(1 \leq n \leq 9 .\) b) Does the truth of \(S(k)\) imply that of \(S(k+1)\) for all \(k \in \mathbf{Z}^{+}\)?

a) Determine all \(w, x, y \in \mathbf{Z}\) that satisfy the following system of Diophantine equations. $$ \begin{aligned} &w+x+y=50 \\ &w+13 x+31 y=116 \end{aligned} $$ b) Is there any solution in part (a) where \(w, x, y>0\) ? c) Is there any solution in part (a) where \(w>10, x>28\), and \(y>-15\) ?

a) How many positive divisors are there for \(n=2^{14} 3^{9} 5^{8} 7^{10} 11^{3} 13^{5} 37^{10}\) ? b) For the divisors in part (a), how many are i) divisible by \(2^{3} 3^{4} 5^{7} 11^{2} 37^{2}\) ? ii) divisible by \(1,166,400,000\) ? iii) perfect squares? iv) perfect squares that are divisible by \(2^{2} 3^{4} 5^{2} 11^{2}\) ? v) perfect squares that are divisible by \(2^{3} 3^{4} 5^{4} 7^{5}\) ? vi) perfect cubes? vii) perfect cubes that are multiples of \(2^{10} 3^{9} 5^{2} 7^{5} 11^{2} 13^{2} 37^{2} ?\) viii) perfect fourth powers? ix) perfect fifth powers? x) perfect squares and perfect cubes?

a) Develop a recursive definition for the addition of \(n\) real numbers \(x_{1}, x_{2}, \ldots, x_{n}\), where \(n \geq 2\). b) For any real numbers \(x_{1}, x_{2}\), and \(x_{3}\), the associative law of addition states that \(x_{1}+\) \(\left(x_{2}+x_{3}\right)=\left(x_{1}+x_{2}\right)+x_{3}\). Prove that if \(n, r \in \mathbf{Z}^{+}\), where \(n \geq 3\) and \(1 \leq r

Suppose that \(t \in \mathbf{Z}^{+}\)and that \(p_{1}, p_{2}, p_{3}, \ldots, p_{t}\) are distinct primes. If \(m \in \mathbf{Z}^{+}\)and the prime factorization of \(m\) is \(p_{1}^{*} p_{2}^{\prime 2} p_{3}^{\prime 3} \cdots p_{t}^{\prime \prime}\), what is the prime factorization (a) for \(m^{2} ?\) (b) for \(m^{3}\) ?
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