Math Is Fun Forum

XxSarahxX

Guest

E-world problem

If anyone could explain how the following is done, it would be greatly appreciated!

In E-World, the only numbers that exist are even integers. An e-prime is an even number that is not divisible by any even numbers. For instance, 6 is an e-prime because it can't be divided by 4 and it can't be divided by 2 in the E-World because 6/2 is 3, which does not exist in this world (and also in E-World, a number is not divisible by itself).

180 has 3 different factorizations as a product of E-primes (6*30, 10*18, and 2*90).  Is 180 the smallest number with 3 factorizations? What is the smallest number with 4 factorizations?

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: E-world problem

Let me try!  24 times 4 is 96. 96/2=48,96/4=24,96/6=16,96/8=12,96/12=8,96/16=6,96/24=4,96/48=2
So that makes four factorizations for 96, since they are paired up.
Let me try again! 16 times 4 is 64.
64/2=32,64/4=16,64/8=8,64/16=4,64/32=2
That makes three factorizations for 64, since two are paired up.
Let me try again! 15 times 4 is 60.  15 is 3x5, hmm.
60/2=30,60/4=15ODD,60/6=10,60/10=6,60/12is5ODD,60/20=ODD,60/30=2
That makes two factorizations for 60.

---

igloo myrtilles fourmis

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: E-world problem

24 times 4 is 96. 96/2=48,96/4=24,96/6=16,96/8=12,96/12=8,96/16=6,96/24=4,96/48=2
So that makes four factorizations for 96, since they are paired up.

They can all be broken down further though, because they're not made of E-primes.
96 only has one E-prime factorisation, which is 2x2x2x2x6. Similarly, 64 is only 2x2x2x2x2x2.

You're alright with 60 though, that can be 2x30 or 6x10.

---

Why did the vector cross the road?
It wanted to be normal.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: E-world problem

An even number n is an E-prime if and only if n ≡ 2 (mod 4) (in the integers). So the first four E-primes are 2, 6, 10, 14. For an even number N to be the product to two E-primes it must be divisible by 4 but not by 8 (in the integers). Hence N = 2×2ׅ – and the trick is to try as many combinations of low odd primes as possible for the product. So smallest number with 4 E-prime factorizations is, I reckon, 420 = 2 × 210 = 6 × 70 = 10 × 42 = 14 × 30.

Last edited by JaneFairfax (2007-09-21 07:20:56)