A prime number problem.?

a and b can be uniquely expressed as products of prime factors

a = a1.a2.a3.....an

b = b1.b2.b3.....bm

ab = (a1.a2.a3.....an)(b1.b2.b3.....bm)

if n|ab then n is some product of the factors from a1....an b1.....bm

let us say we had a composite number n = (n1)(n2)....ni where n1 and n2 are prime factors and there are at least 2 prime factors

we then know that if n|ab then n1 and n2 must both be included in (a1.a2.a3.....an) or (b1.b2.b3.....bm). It can be seen that a and b can always be found such that n1 is a member of {a1.a2.a3.....an} and n2 is a member of {b1.b2.b3.....bm} but n1 is not a member of (b1.b2.b3.....bm) and n2 is not a member of {a1.a2.a3.....an} , or if n1 = n2, they are only a member of each set once. So therefore n can not divide into a or b. For any composite number, we can find a and b such that n|ab but not n|a or n|b

it is trivial that 1|ab and 1|a and 1|b since 1|m is true for all integers m

so we know that the number satisfying the condition is not composite but could be 1.

But does that prove it could also be prime? I am not sure.

If you take a prime n, then it is reduced to one prime factor n, so if n|ab, then n is a member of (a1.a2.a3.....an)(b1.b2.b3.....bm), so it must divide one or both of b, since it must be in one or both sets.

so if n is prime, then it must satisfy the condition, but if it is composite it can't satisfy the condition, if it is one it satisfies the condition.

that should do it

I suspect that a proof by contradiction might work well here. Start by assuming the "if" part:

"Assume that n is a positive integer such that for any positive integers a & b, if n|ab then n|a or n|b."

Then suppose that the "then" part ISN'T true:

"Suppose that n is composite."

Then hopefully you can find a contradiction that results from this, which would mean that n CAN'T be composite, so it has to be 1 or prime.

Possible hint: if n is composite, then it can be written as the product of two other numbers, say, x and y, so n can be replaced with xy.

suppose n is composite, so that n = xy, where x,y > 1.

since n|n, either n|x, or n|y.

since x > 1, n = xy > x. and since y > 1 n = xy > y.

but n cannot divide a positive number less than n.