If x is a positive integer and 50!/30^x is a positive integer, what is

Question

Competition Mode Question

If x is a positive integer and  \frac{50!}{30^x}  is a positive integer, what is the greatest possible value of x ?

A. 11
B. 12
C. 13
D. 14
E. 15

Are You Up For the Challenge: 700 Level Questions

madzaka · Answer

30=2*3*5
Our bottle neck is 5 so we need to know how many 5 are in 50!
we have for every 5 numbers 1 and on 25 and 50 we have 2 fives
thus 10+2=12
if we have 12 fives is 50! we can have 30^12 as the greatest value

Ans (b)

Leadership · Answer

We need to Prime factorizing for 30 -> 2*3*5

So, the maximum number for 2 will be calculated by:
 \frac{50}{2}  +  \frac{50}{2^2}  +  \frac{50}{2^3}  +  \frac{50}{2^4}  +  \frac{50}{2^5}  = 25+12+6+3+1= 47
So, the maximum number for 3 will be calculated by:
 \frac{50}{3}  +  \frac{50}{3^2}  +  \frac{50}{3^3}  = 16+5+1= 22
So, the maximum number for 5 will be calculated by:
 \frac{50}{5}  +  \frac{50}{5^2}  = 10+2 = 12

As 30 is the combination of 2, 3 and 5 - We take the minimum value of them, i.e. - 12. 'B' is the winner

lacktutor · Answer

If x is a positive integer and  \frac{50!}{30^{x} }  is a positive integer, what is the greatest possible value of x ?

30^{x}=2^{x}*3^{x}*5^{x} 
--> In order to get the greatest possible value of x, finding the number of  5s  in  50!  is enough:

+ +...= 10+2 =12  --> the number of  5s  in  50! 
 -->x = 12

The answer is B

freedom128 · Answer

50!/(2^x . 3^x. 5^x)

The greatest value of positive integer x can be determined by following:
 50/5 + 50/(5^2) + 50/(5^3) = 10 + 2 + 0 = 12

FINAL ANSWER IS (B) 12

Abhishek009 · Answer

30 = 2*3*5

Highest value is 5

So, 50!/5 = 10 & 10!/5 = 2

Hence, Answer must be 10 + 2 = 12, Answer must be (B) 12

unraveled · Answer

If x is a positive integer and  \frac{50!}{30^x}  is a positive integer, what is the greatest possible value of x ?

A. 11
B. 12
C. 13
D. 14
E. 15

\frac{50!}{30^x}  =  \frac{50!}{2^x*3^x*5^x}  
Here x would take highest value from  5^x  since  2^x  would give value of x that would falsify the statement.
 \frac{50}{5}  = 10 (highest power of 5 in 50!)
 \frac{50}{25}  = 2 (highest power of 25 in 50!)
10 + 2 = 12
Hence highest power of 5 = 12.

Answer B.

Archit3110 · Answer

30^x = 5*2*3 ; to get highest possible value divide 50!/5 ; 
50!/5 + 50!/25= 10+2; 12
IMO B

If x is a positive integer and 50!/30^x is a positive integer, what is the greatest possible value of x ?

A. 11
B. 12
C. 13
D. 14
E. 15

DavidGMAT1 · Answer

I guess the answer is D.
I think that the exponent must comprise all the 3s that you can find from 1 to 30, which are progressively in the numbers 3,6(3x2),9(3^2),12(3x4),15(3x5),18(3^2x2),21(3x7),24(3x8),27(3^3) and finally 30(3x10).
In total they are 14, hence D.

madgmat2019 · Answer

50!/30^x

30=3*2*5.......so if we can get the no.of 5's in 50! Then that would be sufficient to find X....as 5 is the greatest prime factor of 5

50! Consists of 11 5's

So the max value of X is 11

OA:A

exc4libur · Answer

primefactors(30)=2,3,5
multiples of 5 in 50!: sum quotients 50/5+50/25=10+2=12

Ans (B)

Jawad001 · Answer

30^x = (5*3*6)^x
Greatest possible value of x = 50/5 + 50/25 
 = 10 + 2 = 12
Answer: B

Kinshook · Answer

Asked: If x is a positive integer and  \frac{50!}{30^x}  is a positive integer, what is the greatest possible value of x ?

30 = 2*3*5
Power of 30 will depend on maximum power of 5 in 50!

Power of 5 in 50! = 10 + 2 = 12

IMO B

egmat · Answer

Let's break this down step by step.

We need  50!  divided by  30 ^x to be a positive integer. That means  30 ^x must divide evenly into  50! . We want the LARGEST x possible.

Step 1: Break 30 into primes. 
 30  =  2  ×  3  ×  5 
So  30 ^x =  2 ^x ×  3 ^x ×  5 ^x

This means  50!  must contain AT LEAST x copies of  2 , x copies of  3 , AND x copies of  5 .

Step 2: Count how many times each prime appears in 50!

For 5s  (using the  division shortcut ):
 50  ÷  5  =  10 
 50  ÷  25  =  2 
Total =  12  fives

For 3s: 
 50  ÷  3  =  16 
 50  ÷  9  =  5 
 50  ÷  27  =  1 
Total =  22  threes

For 2s: 
 50  ÷  2  =  25 
 50  ÷  4  =  12 
 50  ÷  8  =  6 
 50  ÷  16  =  3 
 50  ÷  32  =  1 
Total =  47  twos

Step 3: Find the bottleneck. 
We have  47  twos,  22  threes, and  12  fives. Since  30 ^x needs x of EACH prime, x can be at most as large as the SMALLEST count.

The limiting factor is  5 , with only  12  copies.

So the greatest value of x is  12 .

Answer: B

Key Principle:   When finding the highest power of a composite number that divides n!, always factor the base into primes, count each prime separately in n!, and the SMALLEST count is your bottleneck. The rarest prime always limits you — and for most problems, the largest prime factor (here,  5 ) will be the rarest one.

If x is a positive integer and 50!/30^x is a positive integer, what is

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If x is a positive integer and 50!/30^x is a positive integer, what is

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards