Question

What is the highest power of 24 that can divide 80!?
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Answer 1

Answer:

26

Step-by-step explanation:

24 can be written as  2 x 2 x 2 x 3 = 23 x 3

So we will make pairs of 23 x 3 from the number of two’s and number of threes ‘s in 80!

The number of 2’s in 80!  are 78 ;

Number of 8’s in 80 ! is 78/3 = 26

The number of 3’s in 80 ! are 36

So the limiting power in 24  = power of 8 = 26

the highest power of 24 that can divide the 80! is 26.

Answer 2

Answer:

The highest power of 24 that can divide 80! is 26.

Step-by-step explanation:

The factor of 24 = $2^{3}$ × $3$

Highest power of 80! must have common power of  $2^{3}$ × $3$.

So,

$E_{2}$ = Highest power of 2 in 80!

$E_{2}$ = $\frac{80}{2} + \frac{80}{4} + \frac{80}{8} + \frac{80}{16} + \frac{80}{32} + \frac{80}{64}$

    = 40 + 20 + 10 + 5 + 2.5 + 1.25

    = 78.75

For $2^{3}$ ;

Exp $E_{2} ^{'}$  = 78.75 / 3

            = 26

Again,

$E_{3}$ = Highest power of 3 in 80!  

$E_{3}$ = $\frac{80}{3} +\frac{80}{9}+\frac{80}{27}$

    = 26.66 + 8.88 + 2.96

    = 38.5

For $3^{1}$ ;

$E^{'} _{3}$ = 38

Now,

Min ( $E_{2} ^{'}$ , $E^{'} _{3}$ )

Min ( 26 , 38 )

= 26

The highest power of 24 that can divide 80! is 26.