Number of Factors of an Integer
Read this topicExplanations 4
- The number of (positive integer) factors of a positive integer nnn is denoted by d(n)d(n)d(n). It is one of the most important arithmetic functions. There exists a simple formul…
By Kada Williams, Victor Wang, Michael Livshits
- An important concept in understanding numbers is determining the number of positive factors of an integer. It is not only important to know the techniques that can accomplish th…
By Trent Tormoehlen
- Prove that for any real number t>0t>0t>0, there exists a large constant C(t)>0C(t)>0C(t)>0 which satisfies d(n)≤C(t)ntd(n)\le C(t)n^td(n)≤C(t)nt for all positive integers nnn.
By Kada Williams
- For any positive integer ddd, let f(d)f(d)f(d) denote the smallest positive integer with exactly ddd divisors. Let kkk be a natural number. Prove that f(2k)f(2^k)f(2k) divides f…
By Kada Williams
Problems 11
- How many positive integer factors does 848484 have?
- For any positive integer ddd, let f(d)f(d)f(d) denote the smallest positive integer with exactly ddd divisors. Compute f(48)f(48)f(48).
- Let d(n)d(n)d(n) denote the number of (positive integer) factors of a positive integer nnn. Find the smallest solution nnn to the equation d(n3)=5⋅d(n)d(n^3)=5\cdot d(n)d(n3)=5⋅…
- A number is called 6-composite if it has exactly 666 composite factors. What is the 6th smallest 6-composite number? (A number is composite if it has a factor not equal to 111 o…
- There are three flies of negligible size that start at the same position on a circular track with circumference 100010001000 meters. They fly clockwise at speeds of 222, 666, an…
- For any interval A\mathcal{A}A in the real number line not containing zero, define its reciprocal to be the set of numbers of the form 1x\frac 1xx1 where xxx is an element in A…
- The number 393939 has aaa positive integer factors, and the positive integer 424242 has bbb positive integer factors. Find a+ba+ba+b.
- How many positive divisors does 180018001800 have?
- How many positive integers less than 100010001000 have an odd number of factors?
- For any positive integer ddd, let f(d)f(d)f(d) denote the smallest positive integer with exactly ddd divisors. Compute f(9)f(9)f(9).
- How many odd divisors does 30,03030,03030,030 have?