Question

highest power of 72 in 200

Answer 1

Answer:

The Largest power of 72 contained in 200! is 2.

Step-by-step explanation:

Hey Mate,

#Correction in Question : What is the largest power of 72 contained in 200 !

Solution :

200/ 72 + 200/ 72² + 200/ 72³ ........

2.777 + 0.038 + 0.000535 ....

Take only first digit.

2 + 0 + 0 = 2

So,

The Largest power of 72 contained in 200! is 2.

Answer 2

"Answer: 48

The highest power of any number A in any other number B can be determined by first factorizing the number B such that it has one of its factors as A. Now the number of times the number A occurs when B is factorized gives the highest power of A in B

So, in this case when we factorize 200 we get factors as 2, 2, 2, 5, 5 so there is no possibility of getting 72 using some or all of these factors. Hence the highest power of 72 is 0 in the number 200

However, if we are trying to find the highest power of 72 in 200!, then

We need to use the below formula

$\frac{n}{k}+ \frac{n}{k^{2}}+ \frac{n}{k^3}}+..$-----(1)

Where

K is the prime number whose highest power in the factorial of n we are going to determine.

Now this will be the case if $k^{1}$is the highest common factor of n and k

If we need to find the highest power of $k^{a}$ in n! then

$\frac{Highest\quad power\quad of \quad k\quad in\quad n!}{a}$

Step 1: Express 72 in terms of its prime factors$72 = 2 \times 2 \times 2 \times 3 \times 3$

Step 2: Among the prime factors 2 and 3, the highest power of 2 is 200!

Substituting k=2 and n=200 in the formula

We get

$\frac{200}{2}+ \frac{200}{2^{2}}+ \frac{200}{2^{3}}+ \frac{200}{2^{4}}+ \frac{200}{2^5}}+ \frac{200}{2^{6}}+ \frac{200}{2^{7}}+ \frac{200}{2^{8}}$

Taking only the integer value of each fractions we get

100+50+25+12+6+3+1+0 = 197

As here the highest common factor of 72 and a multiple of 2 is 8

$2^{3}=8$

So, the highest power of 8 in 200! Will be the integer value of \$\frac{197}{3} = 65---(1)$

Similarly, for prime factor 3

Substituting k=3 and n=200 in the formula

We get

$\frac{200}{3}+ \frac{200}{3^{2}}+ \frac{200}{3^{3}}+ \frac{200}{3^{4}} =66+ 22+ 7+ 2= 97$

As here the highest common factor of 72 and a multiple of 3 is 9

$3^{2}=9$

So, the highest power of 9 in 200! Will be the integer value of \frac{97}{2} = 48   ---(2)

So, from (1) and (2), we see that the highest power of 9 in 200! Is the lowest. So, the highest power of 72 in 200! Would be the highest power of 9 in 200! = 48  

"