Question

find the highest power of 6 in (30!)​

Answer 1

Answer:

5

Step-by-step explanation:

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

Answer 2

Answer:

Step-by-step explanation:

To find : the highest power of 6 in (30!)​

Concept :

            $E_{p}(n!) = [\frac{n}{p} ] + [\frac{n}{p^2} ] + [\frac{n}{p^3} ] + ... + [\frac{n}{p^k} ]$

where,  p = prime number

            n = a positive integer

            $E_{p}(n!)$  = exponent of p in n!

            [X] = greatest integer <= X

Solution :

​To find the solution we have to get the prime numbers i.e.,

6 = 2 × 3

now using the above formula let,

p = 2, n = 30 then

⇒ 15 +  7 + 3 + 1 = 26

∵  $E_{2}(30!) = 26$ , the highest power of 2 in 30! is 26

now for p = 3,

$E_{3}(30!) = \frac{30}{3} + \frac{30}{3^2} + \frac{30}{3^3} + 0 + 0 + ...$

⇒ 10 + 3 + 1 = 14

∵  $E_{3}(30!) = 14$ , the highest power of 3 in 30! is 14

∵ the highest power of 6 in 30! will be the minimum of 2 and 3 occurrences in 30!

∴ The highest power of 6 in 30! is 14 i.e., $6^{14}$

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