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If the product of the first fifty positive consecutive integers be div 17 Aug 2019, 09:51
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25% (medium)

Question Stats:

80% (00:55) correct 20% (01:09) wrong based on 40 sessions

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If the product of the first fifty positive consecutive integers be divisible by \(7^n\), where n is an integer, then what is the largest possible value of n?

(A) 3

(B) 5

(C) 7

(D) 8

(E) 10

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D
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If the product of the first fifty positive consecutive integers be div 17 Aug 2019, 11:02
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Let the set of numbers from 1 to 50. So we have1×2×...×50=
1×2×...×7×...14×...21×...28×...35×...42×...49×...50.
7=7×1 ,14=7×2 ,21=3×7
28=4×7, 35=5×7 , 42=7×6
49=7×7


7^n= 7^8
Option D

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If the product of the first fifty positive consecutive integers be div 17 Aug 2019, 11:16
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Hovkial
If the product of the first fifty positive consecutive integers be divisible by \(7^n\), where n is an integer, then what is the largest possible value of n?

(A) 3

(B) 5

(C) 7

(D) 8

(E) 10
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Product of the first fifty positive consecutive integers = 50!

The highest factor of 7 is

50/7 = 7
7/7 = 1

So, The highest factor of 7 is 8, Answer must be (D) 8
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If the product of the first fifty positive consecutive integers be div 25 Aug 2019, 05:35
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Hi,

There are two ways to go about it.

FIRST METHOD
Write down all multiples of 7 from 1 to 50 i.e. 7, 14, 21, 28, 35, 42, 49 and then count the number of 7s. Be careful! Few number are product of more than one 7. Don't forget to count them. For example, 49.

Thus, there are eight 7s. So, the answer is D.

SECOND METHOD
Find the largest number divisible by 7 in the list of 1 to 50. It's 49.
7 x 7 = 49

So, there are minimum seven 7s. Eliminate A and B options.

Now, count the numbers which are multiple of 7 in the list of 1 to 7. Only one i.e. 7

So, we have total 8 numbers in all.

Thus, the answer is D.

Please give kudos if you liked the solution. :)


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If the product of the first fifty positive consecutive integers be div 25 Aug 2019, 06:09
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Hovkial
If the product of the first fifty positive consecutive integers be divisible by \(7^n\), where n is an integer, then what is the largest possible value of n?

(A) 3

(B) 5

(C) 7

(D) 8

(E) 10
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Asked: If the product of the first fifty positive consecutive integers be divisible by \(7^n\), where n is an integer, then what is the largest possible value of n?

Largest power of 7 in 50! = 7+1=8

IMO D

Posted from my mobile device

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