The-exponent-of-12-in-100-is-

Question Number 5824 by Bc160200609 last updated on 30/May/16

The exponent of 12 in 100! is

Commented by Yozzii last updated on 30/May/16

12^{48} = 2^{96} 3^{48} b u t I {h a v e n}^{'} t f i g u r e d

o u t y e t h o w t o s h o w t h i s .

T h e i s o l a t i o n o f p o w e r s o f 2 a n d 3

i n t h e p r o d u c t o f 100! i s r e q u i r e d .

12 = 2^{2} \times 3 . T h e r e m a y b e a s i m p l e

m e t h o d \dots

Commented by prakash jain last updated on 30/May/16

Exponent of 2 = ⌊ \frac{100}{2} ⌋ + ⌊ \frac{100}{2^{2}} ⌋ + \dots + ⌊ \frac{100}{2^{6}} ⌋

= 50 + 25 + 12 + 6 + 3 + 1 = 97

Exponent of 3 = ⌊ \frac{100}{3} ⌋ + ⌊ \frac{100}{3^{2}} ⌋ + ⌊ \frac{100}{3^{3}} ⌋ + ⌊ \frac{100}{3^{4}} ⌋

= 33 + 11 + 3 + 1 = 48

12 = 2^{2} \times 3

So highest power of 12 = 48

12^{48} = 2^{96} \times 3^{48}

Commented by Yozzii last updated on 30/May/16

I s t h e r e a n e x p l a n a t i o n f o r t h e

f o r u m u l a ?

Commented by prakash jain last updated on 01/Jun/16

Every multiple of 2 in 1 to 100 adds 1 to

exponent of 100!

Every multiple of 4 in 1 - 100 adds 2 to

exponent of 100!

and so on .

so exponent of 2 in 100!

= ⌊ \frac{100}{2} ⌋ + ⌊ \frac{100}{4} ⌋ + ⌊ \frac{100}{8} ⌋ + ⌊ \frac{100}{16} ⌋ + ⌊ \frac{100}{32} ⌋ + ⌊ \frac{100}{64} ⌋

= 50 + 25 + 12 + 6 + 3 + 1 = 97

Commented by Yozzii last updated on 01/Jun/16

I s e e . T h a n k s!

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