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What is the maximum value of m such that 7^m divides into 14! evenly?

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What is the maximum value of m such that 7^m divides into 14! evenly?  [#permalink]

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New post 05 Mar 2017, 08:25
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What is the maximum value of m such that 7^m divides into 14! evenly?

(A) 1

(B) 2

(C) 3

(D) 4

(E) 5
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Re: What is the maximum value of m such that 7^m divides into 14! evenly?  [#permalink]

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New post 05 Mar 2017, 09:07
vikasp99 wrote:
What is the maximum value of m such that 7^m divides into 14! evenly?

(A) 1

(B) 2

(C) 3

(D) 4

(E) 5


Highest power of 7 that will evenly divide 14! will be -

\(\frac{14}{7} = 2\)

Hence, answer will be (B) 2
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Re: What is the maximum value of m such that 7^m divides into 14! evenly?  [#permalink]

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New post 05 Mar 2017, 10:40
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To calculate the highest power of a prime number in the factorial of a number, we need to do a continued division until the quotient is 0.

Example
What is the highest power of 5 that divides 25!?

First we divide 25 by 5 => the quotient is 5 and the remainder is 0.
Next, we divide the quotient in the previous step by 5. The new quotient is 1 and the remainder is 0.
Next we divide the quotient again by 5. The new quotient is 0 and the remainder is 1.

Now adding all the quotients, we 5+1+0=6.

So the highest power of 5 that divides 25! is 6.

Similarly applying the same method to 14!, we get

Divide 14 by 7 => quotient is 2 and remainder 0
Divide 2 by 7 => quotient is 0 and remainder is 2.

Adding all quotients we get 2+0=2

Highest value of m = 2

Answer B
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Re: What is the maximum value of m such that 7^m divides into 14! evenly?  [#permalink]

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New post 05 Mar 2017, 12:46
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vikasp99 wrote:
What is the maximum value of m such that 7^m divides into 14! evenly?

(A) 1

(B) 2

(C) 3

(D) 4

(E) 5


Translate this problem as 'How many 7s are there in 14!'. The translation is the hard part! But, think of it as a fraction with 7^m in the denominator. That is, the fraction looks like this:

14*13*12*11*...*2*1 / (7*7*7*7*...7) (m times).

You want to know how many of those 7s you can have, and have them still cancel out (resulting in an integer).

You can only cancel out two 7s: one by turning the 14 in the numerator into a 2, and one by crossing out the 7 in the numerator.
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Re: What is the maximum value of m such that 7^m divides into 14! evenly?  [#permalink]

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New post 20 Oct 2018, 11:39
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: What is the maximum value of m such that 7^m divides into 14! evenly? &nbs [#permalink] 20 Oct 2018, 11:39
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