It is easily checked that 677 is prime.
By trying all of 2,3,5,7,11,13,17,19, and 23?
]]>It is easily checked that 677 is prime.
PS: In general:
In the above problem:
]]>This one could be the basis for many others. But like Aurifeuille who used it for n = 14 in 1871 there is much trial and error.
]]>]]>
1.For how many odd positive integers n<1000 does the number of positive divisors of n divide n?
As bobbym pointed out, n must be a perfect square. n=1 is one possibility. For the others, it can be easily checked that all odd perfect squares greater than 1 and less than 1000 are have at most two distinct prime factors in their factorization. Thus the possibilities for n>1 are:
where p and q are distinct primes and a, b positive integers.
First case:
The number of positive divisors of n are
i.e. there are positive divisors. So the possibilites are and . (Not ; that would make n too large.)Second case:
There are only two such
possible, namely and . The number of positive divisors for each number is 9, which does divide each number.Therefore the answer to your question is: There are 5 odd numbers less than 1000 which are divisible by their number of positive divisors, namely 1, 9, 225, 441, and 625.
]]>The answers are these numbers squared.
1, 3, 15, 21, 25 as given above.
3^2 = 9
]]>The answers are these numbers squared.
1, 3, 15, 21, 25 as given above.
]]>1.For how many odd positive integers n<1000 does the number of positive divisors of n divide n?
I'm not following this thread at all.
Let's take n = 3
divisors are {1,3} so the number of them is 2.
2 does not divide 3
Take n = 9
divisors are {1,3,9} That's 3 divisors. 3 divides 9.
I must be misunderstanding something, but I don't know what.
Bob
]]>There is a formula to compute the number of positive divisors of any integer.
those number's divisors have to be odd
Odd numbers have odd divisors.
Even numbers must have one 2 in there prime factorization at least.
]]>