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# Factors of n help

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1. For each integer A such that A is the product of 4 distinct primes
let
1=a<b<c<d<...<p=A
where the lowercase letters are the 16 positive integer divisors of A

Prove if A<1995 then i-h cannot equal 22

I read through the solution for this question and right at the start it said

'Note that 35*57=1995=3*5*7*19.
Suppose that n<1995and i-h=22; then hi=A'

and the rest of the proof relies on that statement

How does n<1995 and i-h=22 make hi=A?

I've shown it true when the largest prime is smaller than the product of the two smaller primes but to go through the rest of the cases would takes ages and isn't very intuitive.
2. (Original post by MathMoFarah)
How does n<1995 and i-h=22 make hi=A?
They seem to assume that h,i are , though not necessarily in that order, and I can see no justification for it either. Where are the four prime factors in ascending order.
3. The problem is that p4 can be larger than p1p2p3, which means they aren't the primes (I assume in which case it would be p1p2p3, p4)
but the number of inequalities which mess with the order are a bit ridiculous :/
4. (Original post by MathMoFarah)
but the number of inequalities which mess with the order are a bit ridiculous :/
Agreed - which makes me think their solution is suspect.

Had to write a programme to confirm their "conjecture"

There may be something in the i-h =22, but I can't see how to utilise it for that initial claim.
5. (Original post by ghostwalker)
Agreed - which makes me think their solution is suspect.

Had to write a programme to confirm their "conjecture"

There may be something in the i-h =22, but I can't see how to utilise it for that initial claim.
Given a factor X | A, A/X is also a factor of A. Surely this implies letters equidistant from the central position will always multiply to equal A?

Am I missing something?
6. (Original post by DFranklin)
Given a factor X | A, A/X is also a factor of A. Surely this implies letters equidistant from the central position will always multiply to equal A?
That's the missing piece - PRSOM.

My number theory is rustier than I imagined.
7. (Original post by ghostwalker)
That's the missing piece - PRSOM.

My number theory is rustier than I imagined.
Well, the "suppose that ..., then hi = A" line kind of leads you in the wrong direction. TBH, the fact that I skimmed the OP probably helped by not sending me up the same garden path...(!)

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