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

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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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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
very hard
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Re: If x is the product of the integers from 1 to 150, inclusive [#permalink]
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]
divyakesharwani wrote:
This is same as finding trailing zero - right.

Yes. The number of trailing zeros is equal to the number of power of 5 in n!.
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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]
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

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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

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

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Re: If x is the product of the integers from 1 to 150, inclusive [#permalink]
The answer must be D, i.e. 5 will have a total power of 37 in 150!.
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If x is the product of the integers from 1 to 150, inclusive [#permalink]
Bunuel EMPOWERgmatRichC
generis

When is it possible to get extra factors of 5 (or other numbers)? I recall doing a similar problem where there is an exception to this rule of just dividing by increasing amounts. I can't find it, but I think there was some extra factors because it involved adding 2 numbers (probably factorials), which added up to have 2 additional factors of 5 or something.
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Re: If x is the product of the integers from 1 to 150, inclusive [#permalink]
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Hi energetics,

In simple terms, you would look for squares, cubes, quads, etc. that divide evenly into the larger product.

For example, in this prompt:
25 = (25)(1) = 5^2
50 = (25)(2) = (5^2)(2)
75 = (25)(3) = (5^2)(3)
...
125 = (25)(5) = (5^2)(5) = 5^3
Etc.

IF a question asks you to combine two numbers (through addition or multiplication), then there will almost always be some additional factors to account for (otherwise, why would the question ask you to do the additional 'math work'?). Remember that NOTHING about a GMAT question is ever 'random' - each question was written by a human to 'test' you on specific concepts, so one of the most important questions you can ever ask yourself when you're working through a Quant or Verbal question is "why was I given this information (because I was given it for some specific reason)?"

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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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x is the product of the integers from 1 to 150, therefore, x = 150!

5y is a factor of x, therefore, 5y fully divides 150!. Finding out the greatest possible value of y means to find the highest power of 5
that fully divides 150!

To find the highest power of a prime number in any factorial, a method of successive division can be employed where we successively divide the quotients till we cannot divide anymore.

The sum of the quotients represents the highest power of the prime number that is contained in a given factorial.

Attachment:

24th June 2021 - Reply 3.JPG [ 55.23 KiB | Viewed 24589 times ]

The image above represents one such calculation for the current question. Dividing 150 by 5, the quotient is 30; dividing the quotient 30 by 5, we get 6; dividing 6 by 5, the quotient is 1 (the remainder can be ignored).

Sum of the quotients = 30 + 6 + 1 = 37.
The highest power of 5 in 150! Is 37 and therefore the greatest possible value for y is 37.

The correct answer option is D.

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