Number-of-even-composite-factors-of-2520-

Question Number 175766 by BaliramKumar last updated on 06/Sep/22

N u m b e r o f e v e n c o m p o s i t e f a c t o r s o f 2520 ?

Commented by mr W last updated on 06/Sep/22

2520 = 2^{3} \times 3^{2} \times 5 \times 7

f o r e v e n f a c t o r s 2 m u s t b e c o n t a i n e d

a t l e a s t o n e t i m e, 2520 m u s t b e e x c l u d e d .

\Rightarrow 3 \times 3 \times 2 \times 2 - 1 = 35

Answered by aleks041103 last updated on 06/Sep/22

l e t s (n) b e t h e n u m b e r o f c o m p o s i t e f a c t o r s

o f n .

l e t n = p_{1}^{r_{1}} p_{2}^{r_{2}} \dots p_{k}^{r_{k}}, w h e r e p_{1, 2, . ., k} a r e p r i m e s

a n d r_{1, 2, \dots, k} \in N = {1, 2, \dots}

a n y f a c t o r f o f n m u s t b e o f t h e f o r m :

f = p_{1}^{t_{1}} p_{2}^{t_{2}} \dots p_{k}^{t_{k}}, w h e r e 0 ⩽ t_{i} ⩽ r_{i} .

\Rightarrow f o r t_{i} w e h a v e r_{i} + 1 p o s s i b i l i t i e s .

You can't use 'macro parameter character #' in math mode

o u t o f t h o s e σ (n) f a c t o r s t h e n o n c o m p o s i t e

a r e 1, p_{1}, p_{2}, \dots, p_{k}, i . e . k + 1 .

\Rightarrow s (n) = σ (n) - k - 1

\Rightarrow s (n) = [\underset{i = 1}{\prod^{k}} (1 + r_{i})] - k - 1

N o w, t h e c o m p o s i t e f a c t o r s a r e 2 t y p e s

d e v i s i b l e b y p_{q} a n d n o t d e v i s i b l e b y p_{q} .

T h e o n e s n o t d e v i s i b l e b y p_{q} a r e a l l t h e

c o m p o s i t e f a c t o r s o f \frac{n}{p_{q}^{r_{q}}}

\Rightarrow T h e a n s w e r i s :

s (n) - s (\frac{n}{p_{q}^{r_{q}}}) = {\begin{cases} [r_{q} \underset{i = 1, i \neq q}{\prod^{k}} (1 + r_{i})] - 1, p_{q} ∣ n \\ [r_{q} \underset{i = 1, i \neq q}{\prod^{k}} (1 + r_{i})], p_{q} ∤ n \end{cases}

i n o u r c a s e :

2520 = 2.5.2.2.7.9 = 2^{3} . 3^{2} . 5.7

\Rightarrow a n s . = 3 (2 + 1) (1 + 1) (1 + 1) - 1 =

= 3.3.2.2 - 1 = 35

\Rightarrow A n s . 35

Commented by aleks041103 last updated on 06/Sep/22

Commented by mr W last updated on 06/Sep/22

2 i s v a l i d, 2520 i n v a l i d ?

Commented by BaliramKumar last updated on 06/Sep/22

t h a n k s

Commented by aleks041103 last updated on 07/Sep/22

W e l l 2 i s e v e n b u t n o t c o m p o s i t e, w h i l e

2520 i s b o t h e v e n a n d c o m p o s i t e a n d a l s o

i s a f a c t o r o f i t s e l f .

S o 2 {d o e s n}^{'} t c o u n t, w h i l e 2520 c o u n t s .