Find the maximum value of n such that 50! is perfectly divisible

Question

Find the maximum value of n such that 50! is perfectly divisible by 12600^n.

(A) 7 
(B) 6
(C) 8 
(D) 5
(E) 9

JeffTargetTestPrep · Accepted Answer

Let’s break 12,600 into prime factors.

12,600 = 126 x 100 = 9 x 7 x 2 x 25 x 4 = 3 x 3 x 7 x 2 x 5 x 5 x 2 x 2 = 2^3 x 3^2 x 5^2 x 7^1

Since 12,600 = 2^3 x 3^2 x 5^2 x 7^1, we really need to determine how many of 5^2 divide into 50!

Let's first determine the number of 5s within 50! by using the following shortcut in which we divide 50 by 5 then divide the quotient of 50/5 by 5, and continue this process until we no longer get a nonzero quotient.

50/5 = 10

10/5 = 2

Since 2/5 does not produce a nonzero quotient, we can stop.

The final step is to add up our quotients; that sum represents the number of factors of 5 within 50!.

Thus, there are 10 + 2 = 12 factors of 5 within 50!.

Since there are 12 factors of 5 in 50!, there are 6 factors of 5^2 = 25 in 50!. Recall that the maximum value of n such that 50! is divisible by 12600^n is determined by the number of 5^2 in 50!. Therefore, n = 6.

Thus, the maximum value of n such that 50!/  = integer is 6.

Answer: B

Harshit401 · Answer

12600 in prime factors==>2^3×3^2×5^2×7
the highest number from 2^3 , 3^2 , 5^2 & 7 is 5^2. So when counting the nuber of 5==>12. and we require 2 5's each time when dividing 50!.So 12 divide by 2 gives 6.

I thnik the answer is 6.

Sent from my Lenovo A6020a46 using  GMAT Club Forum mobile app

mbah191 · Answer

Hi Jeff,
I'm not sure I understand the shortcut. Can you explain why we can do this? Thanks!
-Max

saukrit · Answer

Please refer the image explanation below

TheGuyDangerous · Answer

Prime factorization of 12600 = 2^3*3^2*5^2*7

Thought the biggest number (which would have the smallest power) is 7, let's find the number of powers of the prime factors 7 & 5 in 50!

for 7,

50/7 + 50/49 = 7 + 1 = 8 => 7^8

for 5,

50/5 + 50/25 = 10 + 2 = 12 => 5^12

There's a 5^2 & 7^1 in 12600 so 6 powers of 12600 are possible in 50!

Ans B

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Find the maximum value of n such that 50! is perfectly divisible

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