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Chapter 4: Problem 22

When does a positive integer \(n\) have exactly 15 positive divisors?

Short Answer

Expert verified
A positive integer \(n\) has exactly 15 divisors when \(n\) is of the form \(p^{14}\) or \(p^2*q^4\) where \(p\) and \(q\) are prime numbers.

Step by step solution

01

Prime Factorization

Every positive integer \(n\) can be expressed uniquely as a product of powers of primes (this is the Fundamental theorem of arithmetic). Let's say \(n=p1^{e_1} . p2^{e_2} . p3^{e_3} . . . pk^{e_k}\) is the prime factorization of \(n\), where \(p1, p2 . . . pk\) are distinct primes and \(e_1, e_2 . . . e_k\) are respective powers.
02

Calculate Number of Divisors

The number of divisors of \(n\) is \((e_1 + 1) * (e_2 + 1) * (e_3 + 1) * . . . * (e_k + 1)\). In our case, this has to equal 15. Noting that 15 is itself a prime number and can only be written as a product of 1 and 15 or 3 and 5.
03

Determine Prime Powers

Therefore, there are two possibilities for \(n\) having exactly 15 divisors by arranging the products of prime powers.1. When \(n\) is the 14th power of a prime (i.e., one prime number with an exponent of 14). 2. When \(n\) is the product of a square of a prime and a fourth power of a different prime. (i.e., 2 different prime numbers, one with an exponent of 2 and one with an exponent of 4)
04

Find the Numbers

In case 1, \(n\) can be \(2^{14}\), \(3^{14}\), \(5^{14}\), and so on depending upon the number of prime numbers you want to choose till infinity. In case 2, \(n\) can be \(2^2*3^4\), \(2^2*5^4\), \(2^2*7^4\), \(2^4*3^2\), \(2^4*5^2\), \(3^2*2^4\), \(3^2*5^4\), \(3^4*2^2\), \(3^4*5^2\), and so on depending upon the number of prime numbers you want to choose till infinity.

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Most popular questions from this chapter

Let \(n \in \mathbf{Z}^{+}\)with \(n=r_{k} \cdot 10^{k}+\cdots+r_{2} \cdot 10^{2}+r_{1} \cdot 10+r_{0}\) (the base-10 representation of \(n\) ). Prove that (a) \(2 \mid n\) if and only if \(2 \mid r_{0}\); (b) \(4 \mid n\) if and only if \(4 \mid\left(r_{1} \cdot 10+r_{0}\right)\); and, (c) \(8 \mid n\) if and only if \(8 \mid\left(r_{2} \cdot 10^{2}+r_{1} \cdot 10+r_{0}\right)\). State a general theorem suggested by these results.

Let \(a, b \in \mathbf{Z}^{*}\) where \(a \geq b\). Prove that \(\operatorname{gcd}(a, b)=\operatorname{gcd}(a-b, b)\).

For any \(n \in \mathbf{Z}, n \geq 0\), prove that a) \(2^{2 n+1}+1\) is divisible by \(3 .\) b) \(n^{3}+(n+1)^{3}+(n+2)^{3}\) is divisible by \(9 .\) c) \(\frac{n^{7}}{7}+\frac{n^{3}}{3}+\frac{11 n}{21}\) is an integer.

Find the value of the largest positive integer \(n\) such that \(2^{n}\) divides \(22 !\)

Give a recursive definition for the set of all a) positive even integers. b) nonnegative even integers.
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