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Smallest positive factor PS. Help.

Expert replies
by aman88 » Wed Dec 26, 2012 12:13 am
Both 52 and 33 are factors of n × 25 × 62 × 73 where n is a positive integer. What is the smallest possible positive value of n?

A. 25
B. 27
C. 45
D. 75
E. 125

OA D

Can someone please explain this problem.

Thanks.
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Source: — Problem Solving |

by Anurag@Gurome » Wed Dec 26, 2012 12:25 am
Please check your source.
I think the original question is as follows...
Both 5^2 and 3^3 are factors of n × 2^5 × 6^2 × 7^3 where n is a positive integer. What is the smallest possible positive value of n?
Say, N = n × 2^5 × 6^2 × 7^3

If 5^2 and 3^3 are factors of N, then N must contain two 5s and three 3s.

Now, 2^5 × 6^2 × 7^3 does not contain any 5. But it does contain two 3s in 6^2 = (2*3)^2

Hence, the missing two 5s and one 3 must be a factor of n.
Hence, minimum possible value of n is (5^2)*3 = 3*25 = 75

The correct answer is D.

PS : According to the question you have posted, following the similar logic the minimum possible value of n should be 33*(52/2) = 33*26
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by aman88 » Wed Dec 26, 2012 12:40 am
Yes, you are right sir. I posted the question incorrectly. Sorry for that.

The original question is what you've mentioned as:
"Quote:
Both 5^2 and 3^3 are factors of n X 2^5 X 6^2 X 7^3 where n is a positive integer. What is the smallest possible positive value of n?"

**I have one question though.
What if the question asked the largest possible value of n?

Thanks.
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by Anurag@Gurome » Wed Dec 26, 2012 9:58 pm
aman88 wrote:What if the question asked the largest possible value of n?
There is no definite answer to this question. As our requirement is n must have two 5s and one 3, we can find the minimum positive value for n. But the maximum value of n can be the largest possible positive multiple of 75 which we cannot definitely determine.
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