Let-n-2-31-3-19-how-many-positive-integer-divisors-of-n-2-are-less-than-n-but-do-not-divide-n-

Question Number 7750 by Tawakalitu. last updated on 13/Sep/16

L e t n = (2^{31}) \times (3^{19}) h o w m a n y p o s i t i v e i n t e g e r

d i v i s o r s o f n^{2} a r e l e s s t h a n n b u t d o n o t d i v i d e n

Commented by Yozzia last updated on 13/Sep/16

n^{2} = 2^{62} \times 3^{38}

\Rightarrow t o t a l n u m b e r o f d i v i s o r s o f n^{2} e q u a l s

(62 + 1) (38 + 1) = 63 \times 39 = 2457

A l l d i v i s o r s o f n^{2} t h a t d o n o t d i v i d e n

t a k e t h e f o r m 2^{a} 3^{b} w h e r e 32 ⩽ a ⩽ 62 o r 20 ⩽ b ⩽ 38 .

(i) S u p p o s e 32 ⩽ a ⩽ 62 a n d 0 ⩽ b ⩽ 19 .

W e c a n t h e n w r i t e 2^{a} 3^{b} = 2^{31 + q} 3^{b}

w h e r e 1 ⩽ q ⩽ 31 \Rightarrow t o t a l n u m b e r o f d i v i s o r s

= 31 \times (19 + 1) = 31 \times 20 = 620

(i i) S u p p o s e 32 ⩽ a ⩽ 62 a n d 20 ⩽ b ⩽ 38 .

\Rightarrow 2^{a} 3^{b} = 2^{31 + q} 3^{19 + x} w h e r e 1 ⩽ q ⩽ 31

a n d 1 ⩽ x ⩽ 19 \Rightarrow n u m b e r o f d i v i s o r s

i s 31 \times 19 = 589 .

(i i i) S u p p o s e 0 ⩽ a ⩽ 31 a n d 20 ⩽ b ⩽ 38 .

\Rightarrow N u m b e r o f d i v i s o r s i s (31 + 1) \times 19 = 32 (20 - 1) = 608

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R e q u i r e d t o t a l = 620 + 589 + 608 = 1817

Commented by Tawakalitu. last updated on 14/Sep/16

T h a n k s s o m u c h f o r y o u r t i m e s i r .

Commented by Tawakalitu. last updated on 14/Sep/16

I r e a l l y a p p r e c i a t e .