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If x is the product of the integers from 1 to 150, inclusive

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If x is the product of the integers from 1 to 150, inclusive [#permalink]

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If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y ?

A) 30
B) 34
C) 36
D) 37
E) 39
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Re: If x is the product of the integers from 1 to 150, inclusive [#permalink]

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bgribble wrote:
If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y ?

A) 30
B) 34
C) 36
D) 37
E) 39


\(x=150!\). We need to find the power of 5 in prime factorization of 150!.

150/5 + 150/5^2 + 150/5^3 = 30 + 6 + 1 = 37 (check here: everything-about-factorials-on-the-gmat-85592.html).

Answer: D.
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Re: If x is the product of the integers from 1 to 150, inclusive [#permalink]

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bgribble wrote:
If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y ?

A) 30
B) 34
C) 36
D) 37
E) 39


It basically asks for the number of 5s in 150!

150/5 + 150/25 + 150/125 = 30 + 6 + 1. Hence 37 Option d)
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Re: If x is the product of the integers from 1 to 150, inclusive [#permalink]

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Bunuel wrote:
bgribble wrote:
If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y ?

A) 30
B) 34
C) 36
D) 37
E) 39


\(x=150!\). We need to find the power of 5 in prime factorization of 150!.

150/5 + 150/5^2 + 150/5^3 = 30 + 6 + 1 = 37 (check here: everything-about-factorials-on-the-gmat-85592.html).

Answer: D.


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Re: If x is the product of the integers from 1 to 150, inclusive [#permalink]

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bgribble wrote:
If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y ?

A) 30
B) 34
C) 36
D) 37
E) 39


total number of 5 is 150/5=30
among 30 there are 25 50 75 100 125 150

contain 1,1,1,1 ,2 , 1 the number of 5 more

total 30+7

d
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Re: If x is the product of the integers from 1 to 150, inclusive [#permalink]

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New post 25 Jan 2017, 08:09
1) To paraphrase the question, we need to find all the prime factors 5 of the number 150!
2) 150/5=30; 150/25=6; 150/125=1. The total number of 5's is 30+6+1=37

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Re: If x is the product of the integers from 1 to 150, inclusive [#permalink]

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Re: If x is the product of the integers from 1 to 150, inclusive [#permalink]

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Re: If x is the product of the integers from 1 to 150, inclusive [#permalink]

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5^3 < 150 < 5^4

Hence, the total number of 5 in 150!:

150/5^1 + 150/5^2 + 150/5^3 = 30 + 6 + 1 = 37

So, y = 37
Ans: D.

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Re: If x is the product of the integers from 1 to 150, inclusive [#permalink]

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New post 23 Oct 2017, 08:32
bgribble wrote:
If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y ?

A) 30
B) 34
C) 36
D) 37
E) 39


It is asking the number of 5 when the multiplication is written in terms of prime factors

So, y = [150/5] + [150/25] + [150/125] = 30 + 6 + 1 = 37

Answer D
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If x is the product of the integers from 1 to 150, inclusive [#permalink]

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bgribble wrote:
If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y ?

A) 30
B) 34
C) 36
D) 37
E) 39

Spelled out a little more:

1) \(x\) = product of integers from 1 to 150
\(x\) = 150 * 149* 148 . . .* 3 * 2 *1:
That is, \(x\) = 150!

2) \(5^{y}\) is a factor of 150! What is the greatest possible value of \(y\)?

Using \(\frac{150}{5^{y}}\), consider each power \(y\), of 5. Do not worry about remainders.

\(\frac{150}{5^1}\) = 30
(5 divides into 150 thirty times)

\(\frac{150}{5^2}\) = 6
(25 divides into 150 six times)

\(\frac{150}{5^3}\) = 1
(125 divides into 150 only once. Ignore the remainder.)

\(5^4 = 625\) -- too large to divide into 150 as a factor.

3) Add the results: 30 + 6 + 1 = 37

Answer D

Once you know the theory and method, questions such as this one are pretty straightforward. The stats here might indicate that the suggestion below is indispensable.

Bunuel wrote:
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If x is the product of the integers from 1 to 150, inclusive   [#permalink] 23 Oct 2017, 09:55
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