Question-173431

Question Number 173431 by mnjuly1970 last updated on 11/Jul/22

Commented by Rasheed.Sindhi last updated on 11/Jul/22

(a, b) = (10, 13), (13, 10);

n = 3197 = 23 \times 139

Commented by mr W last updated on 11/Jul/22

v e r y r i g h t s i r!

Answered by mr W last updated on 11/Jul/22

n = a^{3} + b^{3} = (a + b) [{(a + b)}^{2} - 3 a b] = A \times B

A = a + b = 23

B = {(a + b)}^{2} - 3 a b = 23^{2} - 3 a b

n_{m i n} \Rightarrow B_{m i n} \Rightarrow {(a b)}_{m a x}

(a b) ⩽ {(\frac{a + b}{2})}^{2} = {(\frac{23}{2})}^{2}

f o r a, b \in R, {(a b)}_{m a x} i s w h e n a = b = \frac{23}{2} .

f o r a, b \in N, {(a b)}_{m a x} i s w h e n a a n d b

a r e a t m o s t e q u a l t o e a c h o t h e r, b e s i d e s

B = 23^{2} - 3 a b s h o u l d h a v e n o p r i m e

f a c t o r l e s s t h a n 23 .

w e s t a r t w i t h t h e b e s t a = 11, b = 12 :

B = 23^{2} - 3 \times 11 \times 12 = 133 = 7 \times 19 \Rightarrow b a d!

n o w w e t r y t h e n e x t b e s t a = 10, b = 13 :

B = 23^{2} - 3 \times 10 \times 13 = 139 = p r i m e \Rightarrow o k!

b i n g o! t h i s i s a d i r e c t h i t!

\Rightarrow n_{m i n} = 23 \times 139 = 3197

s i m i l a r y f o r n_{m a x} :

w e s t a r t w i t h a = 1, b = 22 :

B = 23^{2} - 3 \times 1 \times 22 = 463 = p r i m e \Rightarrow o k!

\Rightarrow n_{m a x} = 23 \times 463 = 10649

Commented by mnjuly1970 last updated on 12/Jul/22

very nice solution

sir W \dots grateful \dots thanks alot . .

Commented by Rasheed.Sindhi last updated on 12/Jul/22

G reat sir!

Commented by Tawa11 last updated on 13/Jul/22

Great sir

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