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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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New post 20 Oct 2013, 07:13
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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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New post 20 Oct 2013, 07:17
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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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New post 20 Oct 2013, 07:18
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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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New post 20 Oct 2013, 07:18
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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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New post 30 Apr 2015, 00:28
2
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, 09:09
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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New post 23 Oct 2017, 08:11
1
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, 09:32
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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New post 23 Oct 2017, 10:55
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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

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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 14 Mar 2018, 19:11
Hi All,

Since we're multiplying a big string of numbers together, this question comes down to "prime factorization"....we need to "find" all of the 5s that exist in this string of numbers. As a hint, some numbers have MORE THAN one 5 in them.

To start, we know that there are 30 multiples of 5 in the string from 1 to 150, so that's 30 5s right there.

Now, we need to think about numbers that have more than one 5 in them....

5, 10, 15....these all have just one 5

25, 50, 75, 100, 150...these all have TWO 5s; we already counted one of the 5s in each, so we have to now add the other one to the total = +5 more

125....this has THREE 5s; we already counted one of the 5s, so we have to now add the other two to the total = +2 more

30 + 5 + 2 = 37 fives.

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

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New post 26 Mar 2018, 04:01
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 is the product of the integers from 1 to 150, inclusive means x = 150!

5^y is a factor of 150! means, \(\frac{150!}{5^y}\) \(= I\), where "I" is an integer

We need to calculate the no. of 5s in 150!

= \(\frac{150}{5} + \frac{150}{25} + \frac{150}{125}\)

= \(30 + 6 + 1\)

= \(37\)

(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 27 Mar 2018, 11:34
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



The product of the integers from 1 to 150, inclusive, is 150!. To determine the number of factors of 5 within 150!, we can use the following shortcut in which we divide 150 by 5, and then divide the quotient of 150/5 by 5 and continue this process until we can no longer get a nonzero integer as the quotient.

150/5 = 30

30/5 = 6

6/5 = 1 (we can ignore the remainder)

Since 1/5 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 150!.

Thus, there are 30 + 6 + 1 = 37 factors of 5 within 150!.

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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New post 03 Apr 2018, 08:35
The answer must be D, i.e. 5 will have a total power of 37 in 150!.
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Re: If x is the product of the integers from 1 to 150, inclusive &nbs [#permalink] 03 Apr 2018, 08:35
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