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What is the highest power of 3 in the value of X! ?  [#permalink]

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What is the highest power of 3 in the value of x!, where x is a positive integer?

(1) The highest power of 9 in the value of x! is 9.
(2) The highest power of 6 in the value of x! is 19. This question was provided by Experts Global for the Game of Timers Competition _________________
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Re: What is the highest power of 3 in the value of X! ?  [#permalink]

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(1) The highest power of 9 in the value of x! is 9.
(2) The highest power of 6 in the value of x! is 19.

Using statement 1
The value of 9 will account for all 3 that formed groups of 2 such as 3 *3
hence highest value will be 9*2 =18. But there can be some 3 which does not form a pair
hence it could b 19 also. Thus not sufficient.

Using statement 2
highest power of 6 = 2*3
for every 3 there will be a corresponding 2 hence the highest power of 3 will be same as highest power of 6 = 19
Thus B is sufficient

##### General Discussion
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What is the highest power of 3 in the value of X! ?  [#permalink]

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What is the highest power of 3 in the value of x!, where x is a positive integer?

(1) The highest power of 9 in the value of x! is 9.
(2) The highest power of 6 in the value of x! is 19.

From St 1 we just know that 9 is the highest power 9 in x!. So we can have either 18 or 19 units of 3 in x!. For example, I have tried 39! or 42!. In both cases, we have 9 in power of or 9^9. Thus not sufficient

We are told that highest power of 6 in the value of x! is 19. Since we get more 2s than 3s, power of 6 in x! is determined by number of 3s in x!. So, $$\frac{x!}{3}$$+$$\frac{x!}{3^2}$$+$$\frac{x!}{3^3}$$=19. Value 42! fits
So x! is 42!, from here we can count number of 3. Hence, St 2 is sufficient

Originally posted by mira93 on 05 Jul 2019, 08:13.
Last edited by mira93 on 06 Jul 2019, 04:51, edited 4 times in total.
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Re: What is the highest power of 3 in the value of X! ?  [#permalink]

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What is the highest power of 3 in the value of x!, where x is a positive integer?

(1) The highest power of 9 in the value of x! is 9. --> not correct: x! = 9^9 *k = 3^18*k(k is an integer, not multiple of 9, but can be or can't be multiple of 3) where k can be or can't be multiple of 3, so the highest power of 3 can be 18 or 19
(2) The highest power of 6 in the value of x! is 19. --> correct: x! = 6^19*m(m is an integer, not multiple of 3) = 3^19*n, so the highest power of 3 must be 19

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Re: What is the highest power of 3 in the value of X! ?  [#permalink]

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Because as per A 3 can be 18 and 19. in both cases we will get 9 9's.

As per B 3 is 19 only, Hence B is sufficient.
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What is the highest power of 3 in the value of X! ?  [#permalink]

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Statement 1: Highest power of 9 is 9. Which implies highest power of 3 could be 18 or 19.

Statement 2: Highest power of 6 is 19. This implies the highest power of 2 and 3 together is 19. However we don't know the highest power of 3.

Combining both statemens together, the highest power of 3 has to be 19.

Both statements together are required.

Option C.

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Originally posted by prashanths on 05 Jul 2019, 08:23.
Last edited by prashanths on 06 Jul 2019, 01:19, edited 1 time in total.
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imo option B,

condition a, signifies 9 ^9, which says 3^18 or 3^19 both can be answers

condtion b, says 6^19, which is equivalent to 3^19 as no of 2 will be definitely more than 19, to form 6^19
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Re: What is the highest power of 3 in the value of X! ?  [#permalink]

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What is the highest power of 3 in the value of x!, where x is a positive integer?

(1) The highest power of 9 in the value of x! is 9.
(2) The highest power of 6 in the value of x! is 19.

The power of 6 always indicates power of 3 as power of 2 will always be more than power of 3.

For eg: if x=9 then in 9! the highest power of 2 is 7 but highest power of 3 is 4. so highest power of 6 will be 4 which is same as highest power of 3.
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Re: What is the highest power of 3 in the value of X! ?  [#permalink]

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What is the highest power of 3 in the value of x!, where x is a positive integer? B, Statement# 2 only.

(1) The highest power of 9 in the value of x! is 9.
Let's see with powers of 2, In 8! highest power of 2 is 7, but highest power of 4 is 3. So not every time highest power of x^2 is twice that of highest power in x for a given factorial y!. Insufficient.

(2) The highest power of 6 in the value of x! is 19.
In 6 the only power of 3 is 1. So Highest power of 3 will also be 19.
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Re: What is the highest power of 3 in the value of X! ?  [#permalink]

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we know that for the power 3 in x factorial we need to count all 3 present in X!

statement 1 : we know the power of 9 in x! but there could be a 3 that couldn't form a pair to be 9 and was not counted so insufficient
statement 2 : we know the power of 6 in x! therefore we have already accounted for all the 3 as for any factorial we will see the number of 2 in the factorial will be greater than the number 3 so all 3's have formed pair .therefore sufficient

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Re: What is the highest power of 3 in the value of X! ?  [#permalink]

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STATEMENT (1) The highest power of 9 in the value of x! is 9.
that means x! is divisible by $$9^9$$
$$\frac{x!}{9^9}$$ = x!/(3^2)^9

=x!/3^18
x! is divisible by 3^18
and 18 is the highest power of 3 in x!
so SUFFICIENT

STATEMENT (2) The highest power of 6 in the value of x! is 19.
that means x! is divisible by 6^19
x!/6^19 = x!/(2*3)^19

x!/2^19*3^19
x! is divisible by 3^19
and 19 is the highest power of 3 in x!
so SUFFICIENT

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Re: What is the highest power of 3 in the value of X! ?  [#permalink]

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What is the highest power of 3 in the value of x!, where x is a positive integer?

Option 1: The highest power of 9 in the value of x! is 9.

Now in option 1 let us consider 2 possibilities,

1 - Can the highest power of 9 give us the highest power of 3s in x!? - Maybe. If yes, then 9^9 = 3^18 => highest power of 3 being 18 --> this is possible.
2 - Can the highest power of 9 not give us the highest power of 3s in x!? - Maybe. If yes, then we will have atleast 1 more 3 which will ensure highest power of 9 remains 9 and that highest power of 3 is not 18 but rather 19. Imagine this like 9^9 * 3 being a part of the x!.

Hence since both above considered cases are possible, option 1 is not sufficient.

Option 2: The highest power of 6 in the value of x! is 19.

6^19 = 2^19 x 3^19

Now in option 2 let us consider 2 possibilities,

1 - Can the highest power of 6 give us the highest power of 3s in x!? - Yes. Highest power of 6 will always contain the highest power of 3 as every 3 will havea corresponding 2 to form 6 (vice versa may not be true).
2 - Can the highest power of 6 not give us the highest power of 3s in x!? - No. There cannot be a case where we have highest power of 6 in x! and also an extra 3 which does not have a corresponding 2 available to form a 6.

Hence, since both above considered cases are in synch, option 2 is sufficient.
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Re: What is the highest power of 3 in the value of X! ?  [#permalink]

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1. 9 = $$3^2$$ (We should factor it in primes.)
that means we can use the formula:

If p is a prime number so the highest power of p^a in factorial n is given by:
=> (highest power of p in n!)/(a)
=> (highest power of 3 in x!)/(2)
we can use the formula:
=>$$\frac{(x)}{(3)}$$ + $$\frac{(x)}{(3^2)}$$+ ... = 9

Sufficient.

2. 6 = 2*3
The 3 will have less power than 2 in x!.
So, we can use the formula:
=> $$\frac{(x)}{(3)}$$ + $$\frac{(x)}{(3^2)}$$+ ... = 19
We can find the X.
Sufficient.

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Re: What is the highest power of 3 in the value of X! ?  [#permalink]

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We are asked to find the HIGHEST power of 3 in x!

Statement 1 : The highest power of 9 in the value of x! is 9.

9 = 3^2

This is sufficient. Because 9 only consists of 3s.

Statement 2 : The highest power of 6 in the value of x! is 19.

6 = 3*2

This is NOT sufficient.
All we can say is that x has a minimum of 19 3s.
But 3s can be more. For example, if we have 19 2s and 20 3s, ONLY 19 6s are possible.

Having said that, I can't come up with a number right now, where this can be a scenario.
But on the above reasoning, I'm ruling out statement 2.

So, Ans should be (A)
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(1) The highest power of 9 in the value of x! is 9.
--> x! = 9^9*k
Here, k can be a multiple of 3 or a non-multiple of 3
If k is a multiple of 3, k = 3m --> x! = 9^9*3m = 3^18*3m = 3^19m
--> Highest Power of 3 = 19
If k is a non multiple of 3, x! = 9^9*k = 3^18*k
--> Highest Power of 3 = 18

Insufficient

(2) The highest power of 6 in the value of x! is 19.
--> x! = 6^19*p
--> x! = (2*3)^19*p
--> Highest Power of 3 = 19

Sufficient

IMO Option B

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(1) The highest power of 9 in the value of x! is 9. - Given that X!/3^2 is 9. Insufficient as the number of 3's can be 9*2=18 or 19 As you can have one more multiple of 3 which will not provide a 9 in X!.
(2) The highest power of 6 in the value of x! is 19. = Since the prime factorization of 6 will have 3 has the highest prime (limiting factor of arriving at 6, as 2's will be more common than 3's.)directly provides the number of 3's - 19.

IMO B
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Re: What is the highest power of 3 in the value of X! ?  [#permalink]

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D each one is sufficient
A as 3^2 =9 is sufficient to answer
B as 2*3=6 is also sufficient to find the power of 3

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IMO B

#1
The highest power of 9 in the value of x! is 9.
3^18 or 19 possible insufficient
#2The highest power of 6 in the value of x! is 19
or say 2^19*3^19 ; highest power of 3 is 19
sufficient
IMO B

What is the highest power of 3 in the value of x!, where x is a positive integer?

(1) The highest power of 9 in the value of x! is 9.
(2) The highest power of 6 in the value of x! is 19.

Originally posted by Archit3110 on 05 Jul 2019, 08:51.
Last edited by Archit3110 on 06 Jul 2019, 12:49, edited 1 time in total.
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What is the highest power of 3 in the value of X! ?  [#permalink]

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What is the highest power of 3 in the value of x!, where x is a positive integer?

(1) The highest power of 9 in the value of x! is 9.
If the highest power of 9 in the value of x! is 9 then 9^9 is a factor of x! and so 3^18 is also a factor
But no conclusion may be drawn about highest power of 3 since it may be 18 or 19. INSUFFICIENT.
(2) The highest power of 6 in the value of x! is 19.
If the higher power of 6 in x! is 19 then highest power of 3 in x! is also 19. Since highest power of 2 is always larger than higher power of 3 in x!. SUFFICIENT.

Statement 2 alone is sufficient.

IMO B
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Originally posted by Kinshook on 05 Jul 2019, 08:54.
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Re: What is the highest power of 3 in the value of X! ?  [#permalink]

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IMO B

What is the highest power of 3 in the value of x!, where x is a positive integer?

(1) The highest power of 9 in the value of x! is 9.
(2) The highest power of 6 in the value of x! is 19.

St1: It says highest power of 9 is 9, that means highest power of 3 is 18. What if there is another 3 in the next few numbers, which we are missing here because that's not a factor of 9. (for e.g. 27 is a factor of 9, but 30 is not)

St2: It says highest power of 6 is 19, that means highest power of 3 is 19. There is no situation when a factor of 3 will appear and a factor of 2 will not appear. As factor of 3 is always followed by a factor of 3. (for e.g. - 33 is followed by 32, 39 is followed by 38) Re: What is the highest power of 3 in the value of X! ?   [#permalink] 05 Jul 2019, 08:57

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