2.2.3 Integral Divisors

The following are formulas dealing with integral divisors. On all the formulas, it is necessary to prime factorize the number of interest such that: n=p1e1⋅p2e2⋅p3e3⋯pnenn = p_1^{e_1} \cdot p_2^{e_2} \cdot p_3^{e_3} \cdots p_n^{e_n}.

Number of Prime Integral Divisors

Number of prime integral divisors can be found by simply prime factorizing the number, and count how many distinct prime numbers (p1,p2,…p_1, p_2, \dots) you have in its representation.

Example: Find the number of prime integral divisors of 120. 120 = 2^3 \cdot 3 \cdot 5 \Rightarrow \text{# of prime divisors} = 3

Number of Integral Divisors

Number of Integral Divisors=(e1+1)(e2+1)(e3+1)⋯(en+1)\text{Number of Integral Divisors} = (e_1 + 1)(e_2 + 1)(e_3 + 1)\cdots(e_n + 1)

Example: Find the number of integral divisors of 48. 48=24⋅31⇒(4+1)(1+1)=1048 = 2^4 \cdot 3^1 \Rightarrow (4 + 1)(1 + 1) = 10

Sum of the Integral Divisors

P=p1e1+1−1p1−1⋅p2e2+1−1p2−1⋯pnen+1−1pn−1P = \frac{p_1^{e_1+1} - 1}{p_1 - 1} \cdot \frac{p_2^{e_2+1} - 1}{p_2 - 1} \cdots \frac{p_n^{e_n+1} - 1}{p_n - 1}

Example: Find the sum of the integral divisors of 36. 36=22⋅3236 = 2^2 \cdot 3^2 P=23−12−1⋅33−13−1=71⋅262=7⋅13=91P = \frac{2^3 - 1}{2 - 1} \cdot \frac{3^3 - 1}{3 - 1} = \frac{7}{1} \cdot \frac{26}{2} = 7 \cdot 13 = 91

Number of Relatively Prime Integers less than n

ϕ(n)=n(1−1p1)(1−1p2)⋯(1−1pn)\phi(n) = n \left(1 - \frac{1}{p_1}\right)\left(1 - \frac{1}{p_2}\right)\cdots\left(1 - \frac{1}{p_n}\right)

Example: Find the number of relatively prime integers less than 20. 20=22⋅520 = 2^2 \cdot 5 ϕ(20)=20(1−12)(1−15)=20⋅12⋅45=8\phi(20) = 20 \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{5}\right) = 20 \cdot \frac{1}{2} \cdot \frac{4}{5} = 8

Sum of Relatively Prime Integers less than n

P=ϕ(n)×n2P = \phi(n) \times \frac{n}{2}

Example: Find the sum of the relatively prime integers less than 24. 24=23⋅324 = 2^3 \cdot 3 ϕ(24)=24(12)(23)=8\phi(24) = 24 \left(\frac{1}{2}\right)\left(\frac{2}{3}\right) = 8 P=8×242=8⋅12=96P = 8 \times \frac{24}{2} = 8 \cdot 12 = 96

Proper vs Improper Divisors

A proper integral divisor is any positive integral divisor of the number excluding the number itself. Example: Sum of proper integral divisors of 22 = (Sum of all divisors) - 22 = 3×12−22=36−22=143 \times 12 - 22 = 36 - 22 = 14.

Problem Set 2.2.3

30 has how many positive prime integral divisors
36 has how many positive integral divisors
The sum of the positive integral divisors of 42 is
The number of prime factors of 210 is
The number of positive integral divisors of 80 is
The number of positive integral divisors of 24×52^4 \times 5 is
The sum of the distinct prime factors of 75 total
The number of positive integral divisors of 96 is
The number of positive integral divisors of 100 is
The sum of the positive integral divisors 48 is
The sum of the proper positive integral divisors of 24 is
The sum of the positive integral divisors of 28 is
The number of positive integral divisors of 61×32×236^1 \times 3^2 \times 2^3
The sum of the proper positive integral divisors of 30 is
How many positive integral divisors does 81 have
How many positive integral divisors does 144 have
The sum of the positive integral divisors 3×5×73 \times 5 \times 7 is
The number of positive integral divisors of 65×43×216^5 \times 4^3 \times 2^1
The sum of the positive integral divisors of 20 is
The number of positive integral divisors of 24 is
The sum of the positive integral divisors of 28 is
The number of positive integral divisors of 23×34×452^3 \times 3^4 \times 4^5
The number of positive integral divisors of 64 is
The sum of the proper positive integral divisors of 36 is
The number of positive integral divisors of 24×36×5102^4 \times 3^6 \times 5^{10} is
The number of positive integral divisors of 53×32×215^3 \times 3^2 \times 2^1
How many positive integers less than 90 are relatively prime to 90
Sum of the proper positive integral divisors of 18 is
The sum of the positive integers less than 18 that are relatively prime to 18
The number of positive integral divisors of 12×33×2412 \times 3^3 \times 2^4
How many positive integers less than 16×2516 \times 25 are relatively prime to 16×2516 \times 25
How many integers between 30 and 3 are relatively prime to 30
How many positive integer less than 9×89 \times 8 are relatively prime to 9×89 \times 8
How many integers between 1 and 20 are relatively prime to 20
The number of positive integral divisors of 50×54×2350 \times 5^4 \times 2^3
The sum of the positive integral divisors of 48