Question

The highest power of 3 which is contained in 500:
a) 248
b) 241
c) 246
d) 247​

Answer 1

your answer is option no. c which is 246 because it is divisible by 3 by factorisation method..

Answer 2

Correct question: The highest power of 3 which is contained in 500!

a) 248

b) 241

c) 246

d) 247​

Answer:

The highest power of 3, which is contained in 500! is 247 (option d).

Given,

The number 500!.

To Find,

The highest power of 3 which is contained in 500!.

Solution,

The method of finding the highest power in 500! is as follows -

We know that 500! = 500*499*498*497*496*.......*3*2*1.

To find the highest power of 3 in 500! we have to find how many numbers less than 500 have factors exactly $3, 3^{2}=9, 3^{3}=27, 3^{4}=81, 3^{5}=243.$

We can observe that a number less than 500 cannot have a factor $3^{6}=729$ or greater.

There are $[\frac{500}{3}]=[166.67]=166$ numbers less than 500 which have a factor of 3 with a power of exactly 1. ( [ ] represents the "greatest integer function" )

Similarly, there are $[\frac{500}{9}]=[55.56]=55$ numbers less than 500 which have a factor of 3 with a power of exactly 2.

In this way, there are $[\frac{500}{27}]=[18.51]=18$, $[\frac{500}{81}]=[6.17]=6$ and $[\frac{500}{243}]=[2.06]=2$ numbers less than 500 which have a factor of 3 with a power of exactly 3, 4, 5 respectively.

So the total number of powers of 3 present in 500! is $166+55+18+6+2=247$. So, there are 247 powers of 3 present in 500!.

Hence, The highest power of 3, which is contained in 500! is 247 (option d).

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