if-A-a-b-Z-and-6-40-A-2-a-3-b-find-a-b-max-

Question Number 157687 by MathSh last updated on 26/Oct/21

if A; a; b \in Z^{+} and 6 \cdot 40! = A \cdot 2^{\boldsymbol a} \cdot 3^{\boldsymbol b}

find {(a + b)}_{\boldsymbol \max} = ?

Commented by Rasheed.Sindhi last updated on 26/Oct/21

{(a + b)}_{\boldsymbol \max} = 58

Answered by Rasheed.Sindhi last updated on 26/Oct/21

A c c o u n t o f f a c t o r 2 i n 6.40! :

∙ 2, 4, 6, \dots, 40 (c o u n t i n g o n e f a c t o r) 2^{20}

∙ 4, 8, 12, \dots, 40 (c o u n t i n g 2 n d f a c t o r) 2^{10}

∙ 8, 16, \dots, 40 (c o u n t i n g 3 r d f a c t o r) 2^{5}

∙ 16, 32 (c o u n t i n g 4 t h f a c t o r) 2^{2}

∙ 32 (c o u n t i n g 5 t h f a c t o r) 2^{1}

∙ 6 : 2^{1}

M a x i m u m f a c t o r s o f 2 : 2^{39}

A c c o u n t o f f a c t o r 3 i n 6.40! :

∙ 3, 6, 9, \dots 39 (13 n u m b e r s) : 3^{13}

(c o u n t i n g o n l y o n e f a c t o r)

∙ 9, 18, 27, 36 : 3^{4}

(c o u n t i n g s e c o n d f a c t o r o f e a c h)

∙ 27 (c o u n t i n g 3 r d f a c t o r) 3^{1}

∙ 6 : 3^{1}

M a x i m u m n u m b e r o f f a c t o r s : 3^{19}

{(a + b)}_{m a x} = 39 + 19 = 58

Commented by MathSh last updated on 26/Oct/21

Very nice dear \boldsymbol S er, thank you so much

Answered by mr W last updated on 26/Oct/21

⌊ \frac{40}{2} ⌋ + ⌊ \frac{40}{2^{2}} ⌋ + ⌊ \frac{40}{2^{3}} ⌋ + ⌊ \frac{40}{2^{4}} ⌋ + ⌊ \frac{40}{2^{5}} ⌋ = 20 + 10 + 5 + 2 + 1 = 38

\Rightarrow 40! c o n t a i n s 2^{38}

⌊ \frac{40}{3} ⌋ + ⌊ \frac{40}{3^{2}} ⌋ + ⌊ \frac{40}{3^{3}} ⌋ = 13 + 4 + 1 = 18

\Rightarrow 40! c o n t a i n s 3^{18}

6 = 2^{1} \times 3^{1}

\Rightarrow 6 \times 40! = 2^{39} \times 3^{19} \times n

w i t h g c d (n, 2) = 1, g c d (n, 3) = 1

t h e r e f o r e f o r 6 \times 40! = A \times 2^{a} \times 3^{b},

a_{m a x} = 39, b_{m a x} = 19

{(a + b)}_{m a x} = 39 + 19 = 58

Commented by MathSh last updated on 26/Oct/21

perfect dear S er thank you

perfect dear \boldsymbol S er thank you

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