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