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For nonnegative integers x , y , and m , what is the greatest value

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shasadou
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For nonnegative integers x , y , and m , what is the greatest value [#permalink] New post   23 Jan 2016, 02:08
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For nonnegative integers x, y, and m, what is the greatest value of m for which x^m is a factor of y!?

(1) y=x−1

(2) x is a prime number
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C
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chetan2u
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For nonnegative integers x , y , and m , what is the greatest value [#permalink] New post   23 Jan 2016, 02:23
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shasadou wrote:
For nonnegative integers x, y, and m, what is the greatest value of m for which x^m is a factor of y!?

(1) y=x−1

(2) x is a prime number



Hi,
A Good Q..
the Q statement is "For nonnegative integers x, y, and m, what is the greatest value of m for which x^m is a factor of y!?"
What doe sthis mean..
It means the largest power of x that is there in y!..
formula is \(\frac{y}{x} + \frac{y}{x^2}..\) and so on till the fraction \(\frac{y}{x^z}\) becomes less than 1..
x is a prime number or the biggest prime in any integer..

lets see the sentences..
(1) y=x−1
this means \(x^m\) in (x-1)!..
\(\frac{y}{x} + \frac{y}{x^2}..\). and so on means \(\frac{(x-1)!}{x}\)..
if x is prime, answer is 0..
if not it will depend on x..
say x=6, so y=5..
check for 3s in 5! as 3 is the largest prime number in 6..
\(\frac{5}{3}\)=1 so m=1..
different answers
But we do not know if x is prime or what is the largest prime in the integer x..
insuff


(2) x is a prime number
since there is no corelation in y and x, we cannot answer ..
say y is 50 and prime is 5, then it is \(\frac{50}{5}+\frac{50}{25}=12.\).
and say 4 and prime is 3then 4/3=1..
insuff..

combined .
we know that x is prime and y is x-1..
from this it becomes clear that y! or (x-1)! will not have x..
so power of x will be 0, or m=0..
suff
C
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Re: For nonnegative integers x , y , and m , what is the greatest value [#permalink] New post   28 Sep 2016, 00:33
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shasadou wrote:
For nonnegative integers x, y, and m, what is the greatest value of m for which x^m is a factor of y!?

(1) y=x−1

(2) x is a prime number



We need the greatest value of m for which x^m is a factor of y!
We have no relative values of x, y and m given in the question stem.

(1) y=x−1

Think what this means. Say x = 8. Then y = 7
We need greatest m such that 8^m is a factor of 7!
7! has a 2, a 4 and a 6 so it has four 2s. So it can make one 8. So m = 1.
But what if x = 7? Then y = 6
We need greatest m such that 7^m is a factor of 6!. Can we make 7 out of first 6 numbers? No, because 7 is prime. So it means that you can never make 7 out of any other 2 factors. So m = 0
Not sufficient.

(2) x is a prime number
x could be 7 but y could be either 6 or 10 (or infinite other values). 6! will not have any 7s but 10! will.
Not sufficient.

Using both:
This is our second case above. If x = prime, y is prime - 1. All numbers till prime - 1 will not be able to make a single prime.
i.e. our second case above. If x = 7, then y = 6 and all positive integers till 6 will not be able to make a 7. So m will always be 0.

Answer (C)
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BrentGMATPrepNow
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Re: For nonnegative integers x , y , and m , what is the greatest value [#permalink] New post   23 Jan 2016, 14:21
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Aside: Whenever you see the word "non-negative," you should ask yourself "Why didn't they just say POSITIVE?"
The answer to this question is a big hint.
Non-negative is not the same as positive, since ZERO is a non-negative number and zero is not positive.
So, when you see the word "non-negative," be sure to consider the possibility that the number equals zero.

Cheers,
Brent
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Re: For nonnegative integers x , y , and m , what is the greatest value [#permalink] New post   16 Mar 2016, 06:38
chetan2u wrote:
shasadou wrote:
For nonnegative integers x, y, and m, what is the greatest value of m for which x^m is a factor of y!?

(1) y=x−1

(2) x is a prime number



Hi,
A Good Q..
the Q statement is "For nonnegative integers x, y, and m, what is the greatest value of m for which x^m is a factor of y!?"
What doe sthis mean..
It means the largest power of x that is there in y!..
formula is y/x + y/x^2.. and so on till the fraction y/x^z becomes less than 1..
x is a prime number or the biggest prime in any integer..

lets see the sentences..
(1) y=x−1
this means x^m in (x-1)!..
y/x + y/x^2.. and so on means (x-1)!/x..
if x is prime, answer is 0..
if not it will depend on x..
say x=6, so y=5..
check for 3s in 5! as 3 is the largest prime number in 6..
5/3=1 so m=1..
different answers
But we do not know if x is prime or what is the largest prime in the integer x..
insuff


(2) x is a prime number
since there is no corelation in y and x, we cannot answer ..
say y is 50 and prime is 5, then it is 50/5+50/25=12..
and say 4 and prime is 3then 4/3=1..
insuff..

combined .
we know that x is prime and y is x-1..
from this it becomes clear that y! or (x-1)! will not have x..
so power of x will be 0, or m=0..
suff
C



This took me 4 minutes to solve
Any other methods?
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For nonnegative integers x , y , and m , what is the greatest value [#permalink] New post   28 Sep 2016, 00:13
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Is there any alternate way to crack this problem. Bunuel please help. I tried solving it by making cases and took somewhere close to 3 minutes to figure it out.
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Re: For nonnegative integers x , y , and m , what is the greatest value [#permalink] New post   28 Sep 2016, 00:51
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Keats wrote:
Is there any alternate way to crack this problem. Bunuel please help. I tried solving it by making cases and took somewhere close to 3 minutes to figure it out.


This took me only 1 min to solve.

Here is how I approached.

We are given x, y, and m and non negative integers. So, we need to have x,y and m >=0;

We need to find the greatest value of m such that y!/x^m = Integer.

Now Statement 1 : y=x−1. It means x and y are co-primes. Now, I checked the fraction for x=2, y=1, for this m has to be zero. Now I checked for a larger number say x=10; y = 9, for this m could be 0 and 1 but not greater than that. -- INSUFFICIENT.

Statement 2 : x is prime. Nothing about y. Insufficient

Combining : So, for any value of prime, we will always have m = 0 because positive value of m will NEVER make the two co primes divisible.. HENCE I MARKED C.
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Re: For nonnegative integers x , y , and m , what is the greatest value [#permalink] New post   19 Dec 2016, 05:38
Just ensuring that my understanding of the concepts is correct.
We have to find the maximum value of m for which x^m is is a factor of y!

Statement 1
y+1=x
now if y=5 x=4 then m=1
but if
y=10 and x=9 them m=2
NS

Statement 2 x is prime quite not sufficient in itself.

However combining 1 and 2 we have the following situation where x is prime and y is a coprime thus the only factor is 1
so C

I guess I am making a mistake here. Please correct me
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Re: For nonnegative integers x , y , and m , what is the greatest value [#permalink] New post   20 Dec 2016, 08:39
Question asks: how many number x can be formed using elements in y!.
Stat.1 gives x=y+1. So x is not in y!. Can x be constructed from elements of y!? Stat.2 gives that x is prime, thus x can't be built from other numbers. Answer: C.
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Re: For nonnegative integers x , y , and m , what is the greatest value [#permalink] New post   31 Jul 2017, 13:19
Very tricky.

1) You should see X could be 1 and Y is 0!, 0! is 1 (for whatever reason) therefore M could be any number. Also if X=2, Y=1 therefore M =0 So not sufficient
2) If X =2, Y has no restriction. So Y! could also be 2! so M=1. But Y could also be 9! and M could equal 2 as 3^2 is a factor or 9! as 9! contains a 9 which is a factor of 3^2. So this number could be infinite.

In terms of C--We know Y! is always less than X and X is prime. So if Y! is less than X we know the prime is always above 6! A prime number such as 7^M doesn't go into 6! Because 7 is a prime and there are no prime factors of 7 in 6X5X4X3X2X1. Therefore, M always has to be zero because only 1 is a factor. This is now sufficient.
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For nonnegative integers x , y , and m , what is the greatest value [#permalink] New post   04 Sep 2020, 07:25
VeritasKarishma wrote:

We need the greatest value of m for which x^m is a factor of y!
We have no relative values of x, y and m given in the question stem.
.
.
Using both:
This is our second case above. If x = prime, y is prime - 1. All numbers till prime - 1 will not be able to make a single prime.
i.e. our second case above. If x = 7, then y = 6 and all positive integers till 6 will not be able to make a 7. So m will always be 0.

Answer (C)


Using 1: Possible values of m:
x : 1 2 3 4 5 6 7 8 9 10 ..
y : 0 1 2 3 4 5 6 7 8 9 ..
y! : 1 1 2 6 24 120 720 ..
m : 0 1 0 0 0 1 0 1 1 1..

Also, as x and y are co-prime, I see 'm' takes only two values 0 or 1
We can see the 'greatest value' of m for which x^m is a factor of y! is '1'
Thus, A is sufficient

VeritasKarishma, BrentGMATPrepNow, chetan2u: Can you please point the error in my reasoning?
Thanks.
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Re: For nonnegative integers x , y , and m , what is the greatest value [#permalink] New post   19 Dec 2021, 05:52
chetan2u

What if when X = 2. Now 2 is a prime number and for 2^m to be divisible by (2-1)! or 1!, m can take any value. How is C sufficient ?
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Re: For nonnegative integers x , y , and m , what is the greatest value [#permalink] New post   19 Dec 2021, 06:41
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Namangupta1997 wrote:
chetan2u

What if when X = 2. Now 2 is a prime number and for 2^m to be divisible by (2-1)! or 1!, m can take any value. How is C sufficient ?


You are taking the opposite of what is given.
x^m or 2^m has to be factor of 1!
2^m will be a factor of 1! only when m =0 as 2^0=1.
But if m is 1, 2^1 is 2 and will not be a factor of 1!
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Re: For nonnegative integers x , y , and m , what is the greatest value [#permalink] New post   10 Jun 2022, 07:16
chetan2u wrote:
the Q statement is "For nonnegative integers x, y, and m, what is the greatest value of m for which x^m is a factor of y!?"
What doe sthis mean..
It means the largest power of x that is there in y!..
formula is \(\frac{y}{x} + \frac{y}{x^2}..\) and so on till the fraction \(\frac{y}{x^z}\) becomes less than 1..
x is a prime number or the biggest prime in any integer..

Hi chetan2u

Can you please derive how the above formula is linked to this ques? and how can we infer that "x is a prime number or the biggest prime in any integer."?
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Re: For nonnegative integers x , y , and m , what is the greatest value [#permalink] New post   14 Feb 2024, 12:10
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Re: For nonnegative integers x , y , and m , what is the greatest value [#permalink]
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