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amanvermagmat
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A positive integer N has exactly 4 unique positive factors. What is th 21 Mar 2019, 22:21
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55% (hard)

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55% (01:41) correct 45% (01:44) wrong based on 95 sessions

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A positive integer N has exactly 4 unique positive factors. What is the highest prime factor of N?

(1) 5 is a factor of N.
(2) 25 is a factor of N.
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B
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DavidTutorexamPAL
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A positive integer N has exactly 4 unique positive factors. What is th Updated on: 30 Mar 2019, 00:52
Originally posted by DavidTutorexamPAL on 29 Mar 2019, 08:34.
Last edited by DavidTutorexamPAL on 30 Mar 2019, 00:52, edited 1 time in total.
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amanvermagmat
A positive integer N has exactly 4 unique positive factors. What is the highest prime factor of N?

(1) 5 is a factor of N.
(2) 25 is a factor of N.
Show more

As we have very few factors we have very few options so we can just write them out.
This is an Alternative approach.

(1) Then (1,5) are both factors of N. If prime p other than 5 is also a factor, then the 4 factors must be (1,5,p,5p). If p > 5, then the answer to the quesiton is 'p', but as there are many options for p, this is insufficient.

(2) Then (1,5,25) are factors of N. Similarly to the above, choosing a prime p other than 5 gives (1,5,25,p,5p,25p), too much! So the only prime factor of N is 5 and it is (trivially) also the highest prime factor.
Sufficient.

(B) is our answer
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A positive integer N has exactly 4 unique positive factors. What is th 30 Mar 2019, 00:45
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DavidTutorexamPAL
amanvermagmat
A positive integer N has exactly 4 unique positive factors. What is the highest prime factor of N?

(1) 5 is a factor of N.
(2) 25 is a factor of N.

As we have very few factors we have very few options so we can just write them out.
This is an Alternative approach.

(1) Then (1,5) are both factors of N. If prime p other than 5 is also a factor, then the 4 factors must be (1,5,p,5p). If p > 5, then the answer to the quesiton is 'p', but as there are many options for p, this is insufficient.

(2) Then (1,5,25) are factors of N. Similarly to the above, choosing a prime p other than 5 gives (1,5,25,p,5p,25p), too much! So the only prime factor of p is 5 and it is (trivially) also the highest prime factor.
Sufficient.

(B) is our answer
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DavidTutorexamPAL I am still not clear with the explanation as to how p would be only 5 and nothing greater than it. Could you try elaborating?

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A positive integer N has exactly 4 unique positive factors. What is th 30 Mar 2019, 00:51
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sumi747

I am still not clear with the explanation as to how p would be only 5 and nothing greater than it. Could you try elaborating?

Posted from my mobile device
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Hey sumi747,

Because if you had a different prime factor, there would be more than 4 positive factors total.
For example, say 7 was a factor.
Since 25 is a factor, then 7*25=175 is also a factor, and so are 5*7=35 and 7.
So N must have 1,5,25,7,35,175 as factors, meaning it has more than 4 unique factors, a contradiction.
In general, for any prime p other than 5, you will have 1,5,25,p,5p,25p so at least 6 factors. Then the only prime factor of N is 5.
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A positive integer N has exactly 4 unique positive factors. What is th 31 Mar 2019, 16:41
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St.1:
2x5= 10, highest PF= 5
5x7= 35, highest PF= 7
And so on
Insufficient.

Caution: Do not carry out thoughts from Statement 1 to Statement 2.

St.2:
5^2 is a factor of N.
And total factors of N is 4.

In this case, if another PF besides 5 is a part of N, then min total factors of N will become (2+1)*(1+1)= 3*2= 6.

So, for N to have 4 factors, value of N should be 5^3.
Highest PF is 5.
Sufficient

Ans B
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