What is the largest power of 3 contained in 200!

Higest Power of 3 in 200!

=> 200/3 = 66
=> 66/3 = 22
=> 22/3 = 7
=> 7/3 = 2

So, The highest power of 3 is (D) 97

Hi All,

These types of questions don't require a complex math approach - they can be solved with a bit of 'brute force' and logic.

In real basic terms, we're asked to find all of the '3s' in 200!

We can figure out that 200/3 = 66, so we know that there are at least 66 '3s' in 200! While that answer is among the 5 choices, it seems a bit too 'easy', so let's do a bit more work and list out the first few numbers that we know have a '3' in them:

3 = 3x1
6 = 3x2
9 = 3x3

Notice how both 3 and 6 have just one 3 in them, but 9 has TWO 3s (there's an 'extra' 3 that we have to account for). This implies that there are probably other numbers that include 'extra 3s' that we have to figure out:

To find those extra 3s, we have to look at numbers that contain 'powers of 3'...

3^2 = 9
3^3 = 27
3^4 = 81

3^5 = 243, but that's too big (we're only going up to 200). Keep in mind that a multiple of 81 is also a multiple of 9 and 27, so we don't want to count any of those values more than once.

200/9 = 22, so we know that there are at least 22 extra 3s (and certainly more because of the 27 and 81). With the 66 3s that we already have, those 22 extra 3s increase the total to 88. With the other extra 3s, we'll end up with MORE than 88 3s. There's only one answer that fits that logic...

Final Answer:

GMAT assassins aren't born, they're made,
Rich

How did you know to stop at 200/3^4 and what is the theory behind this. I have been trying all of the questions you have on the put on here, but I guess i do not understand the theory behind this. Could you elaborate?

Thanks,
AK

Link to theory was provided above: Check this: everything-about-factorials-on-the-gmat-85592.html

the power of 3 in 200! is
200/3 + 200/3^2 +200/3^3+200/3^4
66+33+7+2
97.
so answer is D.
remember we always take the quotient in expression (200/3 and others) .

Power of any Prime Number in any factorial can be calculated by following understanding

Power of prime x in n! = [n/x] + [n/x^2] + [n/x^3] + [n/x^4] + ... and so on

Where [n/x] is greatest Integer value of (n/x) less than or equal to (n/x)
i.e. [100/3] = [33.33] = 33
i.e. [100/9] = [11.11] = 11 etc.
Where,
[n/x] = No. of Integers that are multiple of x from 1 to n
[n/x^2] = No. of Integers that are multiple of x^2 from 1 to n whose first power has been counted in previous step and second is being counted at this step
[n/x^3] = No. of Integers that are multiple of x^3 from 1 to n whose first two powers have been counted in previous two step and third power is counted at this step
And so on.....

i.e. Power of prime 3 in 200! = [200/3] + [200/3^2] + [200/3^3] + [200/3^4] + ... and so on

i.e. Power of prime 3 in 100! = 66+ 22 + 7 + 2 + 0... and so on = 97

Answer: Option D

Divide 200 by 3, quotient is 66
Divide 66 by 3, quotient is 22
Divide 22 by 3, quotient is 7
Divide 7 by 3, quotient is 2

Add all the quotients 66+22+7+2 = 7=97

D is correct

can you please answer this question

What is the highest power of 3 that divides 13! ?

anuha

[13/3] + [13/3^2] = 4+1 = 5


ALTERNATE

3*6*9*12

power of 3 in 3 = 1
power of 6 in 3 = 1
power of 9 in 3 = 2
power of 12 in 3 = 1

Total power of 3 in 13! = 1+1+2+1 = 5

Asked: What is the largest power of 3 contained in 200!

Power of 3 in 200! = 66 + 22 + 7 + 2 = 97

IMO D

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