What number only has 2 divisors?

The sum of all positive divisors of a positive integer N N is 12. 12.

Does this necessarily mean that N = 6 N=6 because 1 + 2 + 3 + 6 = 12 ? 1 + 2 + 3 + 6 = 12?

Yes No

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2 solutions

Jordan Cahn
Dec 10, 2018

N = 11 N=11 is also possible.

Yes, the title helps a lot.

Joshua Lowrance - 2 years, 6 months ago

All the positive divisors of a positive integer N N includes at least N N and 1 1 as its divisors. Thus,

N + 1 + Sum of the other divisors = 12 N+1+\text{Sum of the other divisors}=12 ' Sum of the other divisors = 11 N \text{Sum of the other divisors}=11-N (Assuming N to be greater than 1, as N only has 1 divisor i.e 1, and clearly 1 is not equal to 12) \text{(Assuming N to be greater than 1, as N only has 1 divisor i.e 1, and clearly 1 is not equal to 12)}

Now, N N has to be less than or equal to 11 and greater than 1 as per our assumption, otherwise, the sum of other divisors will result in a negative number and positive numbers don't add up to something negative. It could be equal to 11 as 0 would mean there are no other divisors. We can now remove 2, 3, 4, 5, 7 as their divisors add less than 12, and 8, 9, 10 as its sum of divisors is greater than 12. Thus only 6, and 11 are possible.

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